D-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTIC

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美国普林斯顿大学PUMaC竞赛介绍

2，美国普林斯顿大学PUMaC终赛竞赛规则:
(1)8人一队(可组2-3队)
(2)选手年龄在20周岁以下。
(3)选手没有参加过全日制大学课程的学习。
(4)参赛队伍可以自愿选择参加A组或B组的内容，其中A组的内容相对较难。两个组将于同一天进行比赛，并且能力测试Power test的内容是相同的，最终分别评奖。
For personal use only in study and research; not for commercial use
PUMaC 2013 will be held on November 16, 2013.
适合人群:希望未来进入美国名校深造、数学基础良好、愿意通过培训进一步提高的同学。
3，美国普林斯顿大学PUMaC终赛竞赛流程:
(1)能力测试(PowerTest):在竞赛日之前各队将得到能力题目，须在竞赛日当天提交。
(2)个人赛(IndividualTest): 参赛者可以从代数(Algebra)、组合(Combinatorics)、几何(Geometry)和数论(NumberTheory)这4个专题中选择其中的2个，时间为每个专题1小时。所选内容需在规定日期前确定。
l第五步:赴美参加PUMaC终赛。
l注:初赛和终赛中，参赛选手请自带不具有收发功能的字典和计算器入场参赛。
三、赛程和赛制
1，初赛和终赛举办时间:
国内初赛时间:2012年7月16日下午2:00-3:10;地点:101中学双榆树分校;
国内集训时间:10次课，每次2小时，主讲PUMaC历年真题;
美国普林斯顿终赛时间:2012年11月20日左右;地点:美国普林斯顿学。
2.有点难度:2.What is the sum ofall primes p such that 7p - 6p + 2 is divisible by43?Note that 73 = 343≡-1 (mod 43)and that 66 = (63)2 ≡ 1 (mod 43). Therefore, forp≡ 0; 1; 2; 3; 4; 5 (mod 6), 7p -6p +2 ≡2; 3; 15; 0; 32; 3 (mod 43). Therefore, if43|7p -6p +2, p ≡ 3 (mod 6). This means that p = 3 is the only solution, so that the sum ofall solutions is

IntroductiontoAbstractAlgebra.SolutionsManual.…

BrochureMore information from /reports/2178437/Introduction to Abstract Algebra. Solutions Manual. 4th EditionDescription:Praise for the Third Edition". . . an expository masterpiece of the highest didactic value that has gained additional attractivity throughthe various improvements . . ." Zentralblatt MATHThe Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to thebasic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readersadvance to abstract theory by presenting concrete examples of induction, number theory, integers modulon, and permutations before the abstract structures are defined. Readers can immediately begin to performcomputations using abstract concepts that are developed in greater detail later in the text.The Fourth Edition features important concepts as well as specialized topics, including:The treatment of nilpotent groups, including the Frattini and Fitting subgroupsSymmetric polynomialsThe proof of the fundamental theorem of algebra using symmetric polynomialsThe proof of Wedderburn's theorem on finite division ringsThe proof of the Wedderburn–Artin theoremThroughout the book, worked examples and real–world problems illustrate concepts and their applications,facilitating a complete understanding for readers regardless of their background in mathematics. A wealthof computational and theoretical exercises, ranging from basic to complex, allows readers to test theircomprehension of the material. In addition, detailed historical notes and biographies of mathematiciansprovide context for and illuminate the discussion of key topics. A solutions manual is also available forreaders who would like access to partial solutions to the book's exercises.Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at theupper–undergraduate and beginning–graduate levels. The book also serves as a valuable reference andself–study tool for practitioners in the fields of engineering, computer science, and applied mathematics.Contents:0 Preliminaries 10.1 Proofs 10.2 Sets 20.3 Mappings 30.4 Equivalences 41 Integers and Permutations 61.1 Induction 61.2 Divisors and Prime Factorization 81.3 Integers Modulo 111.4 Permutations 132 Groups 172.1 Binary Operations 172.2 Groups 192.3 Subgroups 212.4 Cyclic Groups and the Order of an Element 24 2.5 Homomorphisms and Isomorphisms 282.6 Cosets and Lagrange's Theorem 302.7 Groups of Motions and Symmetries 322.8 Normal Subgroups 342.9 Factor Groups 362.10 The Isomorphism Theorem 382.11 An Application to Binary Linear Codes 433 Rings 473.1 Examples and Basic Properties 473.2 Integral Domains and Fields 523.3 Ideals and Factor Rings 553.4 Homomorphisms 593.5 Ordered Integral Domains 624 Polynomials 644.1 Polynomials 644.2 Factorization of Polynomials over a Field 67 4.3 Factor Rings of Polynomials over a Field 70 4.4 Partial Fractions 764.5 Symmetric Polynomials 765 Factorization in Integral Domains 815.1 Irreducibles and Unique Factorization 815.2 Principal Ideal Domains 846 Fields 886.1 Vector Spaces 886.2 Algebraic Extensions 906.3 Splitting Fields 946.4 Finite Fields 966.5 Geometric Constructions 986.7 An Application to Cyclic and BCH Codes 997 Modules over Principal Ideal Domains 1027.1 Modules 1027.2 Modules over a Principal Ideal Domain 1058 p–Groups and the Sylow Theorems 1088.1 Products and Factors 1088.2 Cauchy s Theorem 1118.3 Group Actions 1148.4 The Sylow Theorems 1168.5 Semidirect Products 1188.6 An Application to Combinatorics 1199 Series of Subgroups 1229.1 The Jordan–H¨older Theorem 1229.2 Solvable Groups 1249.3 Nilpotent Groups 12710 Galois Theory 13010.1 Galois Groups and Separability 13010.2 The Main Theorem of Galois Theory 13410.3 Insolvability of Polynomials 13810.4 Cyclotomic Polynomials and Wedderburn's Theorem 14011 Finiteness Conditions for Rings and Modules 14211.1 Wedderburn's Theorem 14211.2 The Wedderburn–Artin Theorem 143Appendices 147Appendix A: Complex Numbers 147Appendix B: Matrix Arithmetic 148Appendix C: Zorn's Lemma 149Ordering:Order Online - /reports/2178437/ Order by Fax - using the form belowOrder by Post - print the order form below and send to Research and Markets,Guinness Centre,Taylors Lane,Dublin 8,Ireland.Fax Order FormTo place an order via fax simply print this form, fill in the information below and fax the completed form to 646-607-1907 (from USA) or +353-1-481-1716 (from Rest of World). 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脂质分子通过自组装合成不同形貌的介孔材料_英文_

第27卷第1期上海第二工业大学学报 V ol.27 No.1 2010年3月 JOURNAL OF SHANGHAI SECOND POLYTECHNIC UNIVERSITY Mar. 2010 文章编号:1001-4543(2010)01-0022-08Mesoporous Silica with Different Morphology Synthesized bySelf-assembling of Lipid MoleculeJIN Hai-ying（School of Environmental Engineering, Shanghai Second Polytechnic University, Shanghai 201209, P. R. China）Abstract: This article describes one approach based upon the molecular self-assembly of lipids (N-acyl amino acid) to formvarious mesoporous microstructured silicas, together with a general approach for the structural analysis by electronmicroscopy. It has been found that，(1) the microstructure of the mesoporous silicas has been changed from helical ribbon(HR) analogous to folded proteins (Protein α/β helix), hollow sphere, disk-like and chiral hexagonal rod with increasingsynthesis temperature from 0 to 10, 15 and to 20 °C; (2) the ammonium group functionalized disordered mesopores in the wallof HR and hollow sphere, and highly ordered straight and chiral 2d-hexagonal mesopores in the disk-like and helical rod havebeen observed, respectively; (3) the helical mesoporous ribbon silica was exclusively left-handed and the 2d-hexagonal chiralmesoporous silica was right-handed excess when l form N-acyl aminoacid has been used as lipid template. Findings couldlead to a new insight into the formation of chiral materials at the molecular level and would facilitate a more efficient andsystematic approach to generating rationalized chiral libraries.Keyword: helical ribbon; vesicle; chiral; mesoporous silica; lamellar; 2d-hexagonal0 IntroductionIt is well known that self-assembly is used to construct microstructure for functioning of living organisms. Living systems are constructed through the hierarchal assembly of these microstructures on many different size scales. It is essential to mimic biomolecular self-assembly techniques to design, engineer, and fabricate inorganic nano- and microstructures for the development of advanced materials such as sensors, catalysts and separation media[1].Carbohydrate amphiphiles and phospholipids are basic building blocks of biomembranes in living systems. Inthe same fundamentals, in aqueous media, the lipid molecules self-assemble into diverse aggregate microstructured morphologies, depending on the molecular shape and solution condition such as lipid concentration, electrolyte concentration, pH, and temperature[2]. Among these morpholopies, vesicles[3], ultra-thin memberanes[4],nano-tubes[5], helical ribbons[5], helical tubes[6] and others[7] have been utilized as template to create nano-sized and microstructured inorganic materials. More advanced materials, techniques and fabrication mechanism must be developed to render these self-assembled nano- and microstructure inorganics.Recently, we discovered a templating route for preparing well-ordered mesoporous silica based on theself-assembly of anionic surfactants and inorganic precursors in the presence of aminosilane or quaternary aminosilane as a co-structure-directing agent (CSDA) [8], which opened up a new possibility for the synthesis of mesoporous materials using lipid anionic amphiphilic molecules. In our previous work, we have synthesized2d-hexagonally ordered chiral mesoporous silica by using N-miristoyl-L-alanine sodium salt (C14-L-AlaS) partially neutralized in HCl/C14-L-AlaS=0.08-0.10 at room temperature and 0 o C, with an aminosilane or a quaternized aminosilane as a CSDA[10－11]. Here, we approached to synthesize the various microstructured mesoporous silicas by controlling the N-acyl aminoacid (N-AAA) packing parameters upon varying synthesis temperature. It is well knownthat many amphiphilic molecules spontaneously self-assemble to produce vesicles, ribbons and tubes with收稿日期: 2009-12-15; 修回日期:2010-01-14作者简介: 靳海英（1976－），女，黑龙江人，博士，主要研究方向为多孔材料，电子邮件：hyjin@。

Monomial orthogonal polynomials of several variables

MONOMIAL ORTHOGONAL POLYNOMIALS OF SEVERALV ARIABLESYUAN XUAbstract.A monomial orthogonal polynomial of several variables is of the form x α−Q α(x )for a multiindex α∈N d +10and it has the least L 2norm among all polynomials of the form x α−P (x ),where P and Q αare polynomials ofdegree less than the total degree of x α.We study monomial orthogonal poly-nomials with respect to the weight function d +1i =1|x i |2κi on the unit sphere S d as well as for the related weight functions on the unit ball and on thestandard simplex.The results include explicit formula,L 2norm,and explicitexpansion in terms of known orthonormal basis.Furthermore,in the case ofκ1=...=κd +1,an explicit basis for symmetric orthogonal polynomials isalso given.1.IntroductionThe purpose of this paper is to study monomial orthogonal polynomials of several variables.Let W be a weight function deﬁned on a set Ωin R d .Let α∈N d 0.The monomial orthogonal polynomials are of the form R α(x )=x α−Q α(x )withQ αbeing a polynomial of degree less than n =|α|:=α1+...+αd ,and it is orthogonal to all polynomials of degree less than n in L 2(W,Ω);in other words,they are orthogonal projections of x αonto the subspace of orthogonal polynomials of degree n .In the case of one variable,such a polynomial is just an orthogonal polynomial normalized with a unit leading coeﬃcient and its explicit formula is known for many classical weight functions.For several variables,there are many linearly independent orthogonal polynomials of the same degree and the explicit formula of R αis not immediately known.Let Πd n denote the space of polynomials of degree at most n in d variables.The polynomial R αcan be considered as the error of the best approximation of x αbypolynomials from Πd n −1,n =|α|,in L 2(W,Ω).Indeed,a standard Hilbert spaceargument shows thatR α 2= x α−Q α 2=inf P { x α−P 2,P ∈Πd n −1,n =|α|},where · 2is the L 2(W,Ω)norm.In other words,R αhas the least L 2norm among all polynomials of the form x α−P ,where P ∈Πd n −1.Let dωbe the surface measure on the unit sphere S d ={x : x =1},where x is the Euclidean norm of x ∈R d +1.In the present paper we consider the monomial Date :Sept.2,2002,revised Nov.15,2004.1991Mathematics Subject Classiﬁcation.33C50,42C10.Key words and phrases.h -harmonics,orthogonal polynomials of several variables,best L 2approximation,symmetric orthogonal polynomials.Work supported in part by the National Science Foundation under Grant DMS-0201669.12YUAN XU orthogonal polynomials in L2(h2κdω,S d),where(1.1)hκ(x)=d+1i=1|x i|κi,κi≥0,x∈R d+1.The homogeneous orthogonal polynomials with respect to this weight function are called h-harmonics;they are the simplest examples of the h-harmonics associated with the reﬂection groups(see,for example,[4,5,7]and the references therein).The weight function in(1.1)is invariant under the group Z d+12.Let P d+1ndenotethe space of homogeneous polynomials of degree n in d+1variables.The monomial homogeneous polynomials are of the form Rα(x)=xα− x 2Qα(x),where Qα∈P d+1 n−2and n=|α|.In this case,we deﬁne Rαthrough a generating function andderive their various ing a correspondence between the h-harmonics and orthogonal polynomials on the unit ball B d={x: x ≤1}of R d,this also gives the monomial orthogonal polynomials with respect to the weight function(1.2)W Bκ(x)=di=1|x i|2κi(1− x 2)κd+1−1/2,x∈B d,κi≥0.In the caseκi=0for1≤i≤d andκd+1=µ,the weight function W Bκis the classical weight function(1− x 2)µ−1/2for which the monomial polynomials are known already to Hermite(in special cases);see[8,Vol.2,Chapt12].There is also a correspondence between the h-harmonics and orthogonal polynomials on the simplex T d={x:x i≥0,1−|x|≥0}of R d,where|x|=x1+...+x d,which allows us to derive properties of the monomial orthogonal polynomials with respect to the weight function(1.3)W Tκ(x)=di=1|x i|κi−1/2(1−|x|)κd+1−1/2,x∈T d,κi≥0.For these families of the weight functions,we will deﬁne the monomial orthogo-nal polynomials using generating functions,and give explicit formulae for these polynomials in the next section.Ifκ1=...=κd+1,then the weight function is invariant under the action of the symmetric group.We can consider the subspace of h-harmonics invariant under the symmetric group.Recently,in[6],Dunkl gave an explicit basis in terms of monomial symmetric polynomials.Another explicit basis can be derived from the explicit formulae of Rα,which we give in Section3.Various explicit bases of orthogonal polynomials for the above weight functions have appeared in[7,11,16],some can be traced back to[2,8]in special cases. Our emphasis is on the monomial bases and explicit computation of the L2norm. The L2norms or the monomial orthogonal polynomials give the error of the best approximation to monomials by polynomials of lower degrees.We compute the norms in Section4.They are expressed as integrals of the product of the Jacobi or Gegenbauer polynomials.We mention two special cases of our general results,in which P n(t)denotes the Legendre polynomial of degree n:Theorem1.1.Forα∈N d0,let n=|α|.ThenminQ∈Πdn−11vol BB d|xα−Q(x)|2dx=dα!2(d/2)n1di=1Pαi(t)t n+d−1dt,MONOMIAL ORTHOGONAL POLYNOMIALS3 where vol B d=πd/2/Γ(d/2+1)is the volume of B d,andminQ∈Πdn−11d!T d|xα−Q(x)|2dx=dα!2(d)2n1di=1Pαi(2r−1)r n+d−1dr.As the best approximation to xα,the monomial orthogonal polynomials with respect to the unit weight function(Lebesgue measure)on B2have been studied recently in[3].Let us also mention[1],in which certain invariant polynomials with the least L p norm on S d are studied.For h-harmonics,the set{Rα:|α|=n}contains an orthogonal basis of h-harmonics of degree n but the set itself is not a basis.In general,two monomial orthogonal polynomials of the same degree are not orthogonal to each other.On the other hand,for each of the three families of the weight functions,an orthonormal basis can be given in terms of the Jacobi polynomials or the Gegenbauer polynomi-als.We will derive an explicit expansion of Rαin terms of this orthonormal basis in Section5,the coeﬃcients of the expansion are given in terms of Hahn polynomials of several variables.Finally in Section6,we discuss another property of the polynomials deﬁned by the generating function.It leads to an expansion of monomials in terms of monomial orthogonal polynomials.2.Monomial orthogonal polynomialsThroughout this paper we use the standard multiindex notation.Forα∈N m0we write|α|=α1+...+αm.Forα,β∈N m0we also writeα!=α1!···αm!and(α)β=(α1)β1···(αm)βm,where(a)n=a(a+1)...(a+n−1)is the Pochhammer symbol.Furthermore,forα∈N m and a,b∈R,we write aα+b1=(aα1+b,...,aαm+b) and denote1:=(1,1,...,1).Forα,β∈N m0,the inequalityα≤βmeans that αi≤βi for1≤i≤m.2.1.Monomial h-harmonics.First we recall relevant part of the theory of h-harmonics;see[4,5,7]and the reference therein.We shall restrict ourself to the case of hκdeﬁned in(1.1);see also[16].Let H d+1n (h2κ)denote the space of homogeneous orthogonal polynomials of degreen with respect to h2κdωon S d.If allκi=0,then H d+1n (h2κ)is just the space of theordinary harmonics.It is known thatdim H d+1n (h2κ)=dim P d+1n−dim P d+1n−2=n+dd−n+d−2d.The elements of H d+1n (h2κ)are called h-harmonics since they can be deﬁned throughan analog of Laplacian operator.The essential ingredient is Dunkl’s operators, which are a family ofﬁrst order diﬀerential-diﬀerence operators deﬁned by(2.1)D i f(x)=∂i f(x)+κif(x)−f(x1,...,−x i,...,x d+1)i,1≤i≤d+1. These operators commute;that is,D i D j=D j D i.The h-Laplacian is deﬁned by∆h=D21+...+D2d+1.Then∆h P=0,P∈P d+1nif and only if P∈H d+1n(h2κ).Thestructure of the h-harmonics and that of ordinary harmonic polynomials are paral-lel.Some of the properties of h-harmonics can be expressed using the intertwining4YUAN XUoperator ,V κ,which is a linear operator that acts between ordinary harmonics and h -harmonics.It is uniquely determined by the propertiesD i V κ=V κ∂i ,V κ1=1,V κP d +1n ⊂P d +1n .For the weight function h κin (1.1),V κis an integral operator deﬁned by(2.2)V κf (x )= [−1,1]d +1f (x 1t 1,...,x d +1t d +1)d +1 i =1c κi (1+t i )(1−t 2i )κi −1dt,where c µ=Γ(µ+1/2)/(√πΓ(µ)).If any one of κi =0,the formula holds underthe limit (2.3)lim µ→0c µ 1−1f (t )(1−t 2)µ−1dµ=[f (1)+f (−1)]/2.The Poisson kernel,or reproducing kernel,P (h 2κ;x,y )of the h -harmonics is deﬁned by the property (2.4)f (x )=c h S d f (y )P (h 2κ;x,y )f (y )h 2κ(y )dω(y ),c h =Γ(|κ|+d +12)2 i =1Γ(κi +12),for f ∈H d n (h 2κ)and y ≤1,where c h is thenormalization constant of the weight function h 2κon the unit sphere S d ,c h S d h 2κdω=1and dωis the surface measure.Using the intertwining operator,the Poisson kernel of the h -harmonics can be written as P (h 2κ;x,y )=V κ 1− y 2(1−2 y,· + y 2)ρ+1 (x ),ρ=|κ|+d −12,for y <1= x .If all κi =0,then V κ=id is the identity operator and P (h 20;x,y )is the classical Poisson kernel,which is related to the Poisson kernel of the Gegenbauer polynomials1−r 2(1−2rt +r 2)λ+1=∞ n =0n +λλC λn (t )r n .The above function can be viewed as a generating function for the Gegenbauerpolynomials C λn (t ).The usual generating function of C λn ,however,takes the fol-lowing form:(1−2rt +r 2)−λ= ∞n =0C λn (t )r n .Our deﬁnition of the monomialorthogonal polynomials is the analog of the generating function of C λn in severalvariables.Deﬁnition 2.1.Let ρ=|κ|+d −12>0.Deﬁne polynomials R α(x )by V κ 1(1−2 b,· + b 2 x 2)ρ (x )= α∈N d +10b α R α(x ),x ∈R d +1.Let F B be the Lauricella hypergeometric series of type B ,which generalizes the hypergeometric function 2F 1to several variables (cf.[10]),F B (α,β;c ;x )= γ(α)γ(β)γ|γ|x γ,α,β∈N d +10,c ∈R max 1≤i ≤d +1|x i |<1,where the summation is taken over γ∈N d +10.We derive properties of R αin the following.MONOMIAL ORTHOGONAL POLYNOMIALS5 Proposition2.2.The polynomials Rαsatisfy the following properties:(1) Rα∈P d+1nandRα(x)=2|α|(ρ)|α|α!γ(−α/2)γ(−(α+1)/2)γ(−|α|−ρ+1)|γ|γ!x 2|γ|Vκ(xα−2γ),where the series terminates as the summation is over allγsuch that2γ≤α;(2) Rα∈H d+1n(h2κ)and Rα(x)=2|α|(ρ)|α|α!Vκ[Sα(·)](x)for x =1,whereSα(y)=yαF B−α2,−α+12;−|α|−ρ+1;1y21,...,1y2d+1.Furthermore,|α|=nbα Rα(x)=ρn+ρP n(h2κ;b,x), x =1,where P n(h2κ;y,x)is the reproducing kernel of H d+1n(h2κ)in L2(h2κ,S d−1). ing the multinomial and binomial formula,we write(1−2 a,y + a 2)−ρ=(1−a1(2y1−a1)−···−a d(2y d+1−a d+1))−ρ=β(ρ)|β|β!aβ(2y1−a1)β1...(2y d+1−a d+1)βd+1=β(ρ)|β|β!γ(−β1)γ1...(−βd+1)γd+1γ!2|β|−|γ|yβ−γaγ+β.Changing summation indicesβi+γi=αi and using the formulae(ρ)m−k=(−1)k(ρ)m(1−ρ−m)kand(−m+k)k(m−k)!=(−1)k(−m)2km!as well as2−2k(−m)2k=(−m/2)k((1−m)/2)k,we can rewrite the formula as(1−2 a,y + a 2)−ρ=αaα2|α|(ρ)|α|α!γ(−α/2)γ((−α+1)/2)γ(−|α|−ρ+1)|γ|γ!yα−2γ=aα2|α|(ρ)|α|α!yαF B−α2,1−α2;−|α|−ρ+1;1y21,...,1y2d,Using theﬁrst equal sign of the expansion with the function (1−2 b,y + x 2 b 2)−ρ=(1−2 x b,y/ x +x b2)−ρand applying V with respect to y gives the expression of Rαin the part(1).If x =1,then the second equal sign gives the expression of Rαin the part(2).We still need to show that Rα∈H d+1n(h2κ).Let x =1.For y ≤1the generating function of the Gegenbauer polynomials gives(1−2 b,y + b 2)−ρ=(1−2 b b/ b ,y + b 2)−ρ=∞n=0b n Cρn( b/ b ,y ).Consequently,applying Vκon y in the above equation gives|α|=nbα Rα(x)= b n Vκ[Cρn( b/ b ,· )](x), x =1.6YUAN XUOn the other hand,it is known that the reproducing kernel P n(h2κ;x,y)of H d+1n (h2κ)is given by[7,p.190]P n(h2κ;x,y)=n+ρρy n Vκ[Cρn( y/ y ,· )](x), y ≤ x =1,so that|α|=nbα Rα(x)is a constant multiple of P n(h2κ;x,b).Consequently,forany b,bα Rα(x)is an element in H d+1n(h2κ);therefore,so is Rα.In the following let[x]denote the integer part of x.We also use[α/2]to denote ([α1/2],...,[αd+1/2])forα∈N d+1.Proposition2.3.Letρ=|κ|+(d−1)/2.ThenRα(x)=2|α|(ρ)|α|α!(1/2)α−β(κ+1/2)α−βRα(x),whereβ=α−α+12andRα(x)=xαF B−β,−α+β−κ+12;−|α|−ρ+1;x 2x21,...,x 2x2d+1.Proof.By considering m being even or odd,it is easy to verify thatcκ1−1t m−2k(1+t)(1−t2)κ−1dt=(12)[m+12](κ+12)[m+12](−[m+12]−κ+12)k(−[m+12]+12)kforκ≥0.Hence,using the explicit formula of Vκ,the formula of Rαin(1)of Proposition2.2becomes,Rα(x)=2|α|(ρ)|α|α!(1/2)[α+12] (κ+1/2)[α+12]×γ(−α/2)γ((−α+1)/2)γ(−|α|−ρ+1)|γ|γ!(−[(α+1)/2]−κ+1/2)γ(−[(α+1)/2]+1/2)γx 2|γ|xα−2γ.Using the fact that−α2γ−α+12γ=−α+α+12γ−α+12+12γ,the above expression of Rαcan be written in terms of F B as stated.Note that the F B function in the proposition is aﬁnite series,since(−n)m=0if m>n.In particular,this shows that Rα(x)is the monomial orthogonal polynomial of the form Rα(x)=xα− x 2Qα(x),where Qα∈P d n−2.Another generalization of the hypergeometric series2F1to several variables is the Lauricella function of type A,deﬁned by(cf.[10])F A(c,α;β;x)=γ(c)|γ|(α)γ(β)γγ!xγ,α,β∈N d+1,c∈R,where the summation is taken overγ∈N d+10.If all components ofαare even,thenwe can write Rαusing F A.MONOMIAL ORTHOGONAL POLYNOMIALS7Proposition2.4.Letβ∈N d+1.ThenR2β(x)=(−1)|β|(κ+1/2)β(n+ρ)|β|x |2β|F A−β,|β|+ρ;κ+12;x21x 2,...,x2d+1x 2.Proof.Forα=2βthe formula in terms of F B becomesR2β(x)=γ≤β(−β)γ(−β−κ+1/2)γ(−2|β|−ρ+1)|γ|γ!x 2|γ|x2β−2γ,whereγ≤βmeansγ1<β1,...,γd+1<βd+1;note that(−β)γ=0ifγ>β. Changing the summation index byγi→βi−γi and using the formula(a)n−m= (−1)m(a)n/(1−n−a)m to rewrite the Pochhammer symbols,for example,(κ+1/2)β−γ=(−1)γ(κ+1/2)β(−β−κ+1/2)γ,(β−γ)!=(1)β−γ=(−1)|γ|β!(−β)γ,we can rewrite the summation into the stated formula in F A.Let proj n:P d+1n →H d+1n(h2κ)denote the projection operator of polynomialsin P d+1n onto H d+1n(h2κ).It follows that Rαis the orthogonal projection of themonomial xα.Recall that D i is the Dunkl operator deﬁned in(2.1).We deﬁne Dα=Dα11···Dαd+1d+1forα∈N d+1.Let e i=(0,...,0,1,0,...,0)denote the standard basisof R d+1.Proposition2.5.The polynomials Rαsatisfy the following properties:(1)Rα(x)=proj n xα,n=|α|,andRα(x)=(−1)n2n(ρ)nx 2ρ+2n Dαx −2ρ,ρ=|κ|+d−12.(2)Rαsatisﬁes the relationx 2D i Rα(x)=−2(n+ρ)[Rα+ei(x)−x i Rα(x)].(3)The set{Rα:|α|=n,αd+1=0,1}is a basis of H d+1n(h2κ).Proof.Since Rα∈H d+1n (h2κ)and Rα(x)=xα− x 2Q(x),where Q∈P d+1n−2,itfollows that Rα(x)=proj n xα.On the other hand,it is shown in[19]that thepolynomials Hα,deﬁned byHα(x)= x 2ρ+2n Dα x −2ρ,satisfy the relation Hα(x)=(−1)n2n(ρ)n proj|α|xα,from which the explicit formula in the part(1)follows.The polynomials Hαsatisfy the recursive relationHα+ei(x)=−(2|α|+2ρ)x i Hα(x)+ x 2D i Hα(x),which gives the relation in part(2).Finally,it is proved in[19]that{Hα:|α|=n,αd+1=0,1}is a basis of H d+1n(h2κ).In the case ofα=ne i,Rαtakes a simple form.Indeed,let C(λ,µ)n(t)denote the generalized Gegenbauer polynomials deﬁned byC(λ,µ) n (x)=cµ1−1Cλn(xt)(1+t)(1−t2)µ−1dt.These polynomials are orthogonal with respect to the weight function wλ,µ(t)= |t|2µ(1−t2)λ−1/2on[−1,1]and they become Gegenbauer polynomials whenµ=08YUAN XU (use (2.3));that is,C (λ,0)n (t )=C λn (t ).In terms of the Jacobi polynomials P (a,b )n(t ),the generalized Gegenbauer polynomials can be written as C (λ,µ)2n(x )=(λ+µ)n µ+12 nP (λ−1/2,µ−1/2)n (2x 2−1),C (λ,µ)2n +1(x )=(λ+µ)n +1 µ+12 n +1xP (λ−1/2,µ+1/2)n (2x 2−1).(2.5)Recall that the Jacobi polynomial P (a,b )n (t )can be written as a 2F 1function(2.6)P (a,b )n (t )=(a +1)n n !2F 1 −n,n +a +b +1a +1;1−t 2 .For α=ne i ,the F B formula of R αbecomes a single sum since (−m )j =0if m <j ,which can be written in terms of 2F 1.For example,if n =2m +1,thenR (2m +1)e 1(x )=m j =0(−m )j (−m −κ1−1/2)j (−2m −1−ρ+1)j j ! x 2j x 2m −2j +11=x 2m +112F 1 −m,−m −κ1−1/2−2m −1−ρ+1; x 2x 21.This can be written in terms of Jacobi polynomials (2.6),upon changing the sum-mation index by j →m −j ,and further in the generalized Gegenbauer polynomials using (2.5).The result isCorollary 2.6.Let n ∈N 0.Then R ne i (x )=proj n x n i satisﬁesR ne i (x )= x n k (ρ−κi ,κi )n −1C (ρ−κi ,κi )n (x i x ),where k (λ,µ)ndenote the leading coeﬃcient of C (λ,µ)n (t )given by k (λ,µ)2n =(λ+µ)2n µ+12 n n !and k (λ,µ)2n +1=(λ+µ)2n +1 µ+12 n +1n !.(2.7)In the case of ordinary harmonics,that is,κi =0,the polynomials R ne i are given in terms of the Gegenbauer polynomials.2.2.Monomial orthogonal polynomials on the unit ball.The h -spherical harmonics associated to (1.1)are closely related to orthogonal polynomials asso-ciated to the weight functions W B κin (1.2).In fact,if Y ∈H d +1n (h 2κ)is an h -harmonic associated with h κ(y )= d +1i =1|y i |κi that is even in its (d +1)-th variable,Y (y ,y d +1)=Y (y ,−y d +1),then the polynomial P αdeﬁned by(2.8)Y (y )=r n P (x ),y =r (x,x d +1),r = y ,(x,x d +1)∈S d ,is an orthogonal polynomials with respect to W B κ.Moreover,this deﬁnes an one-to-one correspondence between the two sets of polynomials ([17]).Working with polynomials on B d ,the monomials are x αwith α∈N d 0,insteadof N d +10.Since x 2d +1=1− x 2for (x,x d +1)∈S d ,we only consider R αin Deﬁni-tion 2.1with α=(α1,...,αd ,0).The correspondence (2.8)leads to the following deﬁnition:MONOMIAL ORTHOGONAL POLYNOMIALS9Deﬁnition2.7.Letρ=|κ|+d−12.Deﬁne polynomials R Bα(x),α∈N d0,bycκ[−1,1]d1(1−2(b1x1t1+...+b d x d t d)+ b 2)ρdi=1(1+t i)(1−t2i)κi−1dt=α∈N dbα R Bα(x),x∈B d.The polynomials R Bαform a basis of the subspace of orthogonal polynomials of degree n with respect to W Bκ.It is given by the explicit formulaProposition2.8.Letρ=|κ|+(d−1)/2.Forα∈N d0and x∈R d,R B α(x)=2|α|(ρ)|α|α!(1/2)α−β(κ+1/2)α−βR Bα(x),whereβ=α−α+12,andRα(x)=xαF B−β,−α+β−κ+12;−|α|−ρ+1;1x21,...,1x2d.In particular,R Bα(x)=xα−Qα(x),Qα∈Πd n−1,is the monomial orthogonal polynomial with respect to W Bκon B d.Proof.Setting b d+1=0and x =1in the generating function(2.1)shows that the generating function of R Bαis the same as the one for R(β,0)(x).Consequently, R Bα(x)= R(α,0)(x,x d+1)for(x,x d+1)∈S d.Since R(α,0)(x,x d+1)is even in its d+1 variable,the correspondence(2.8)shows that R Bαis orthogonal and its properties can be derived from those of Rα.In particular,ifκi=0for i=1,...,d andκd+1=µso that W Bκbecomes the classical weight function(1− x 2)µ−1/2,then the limit relation(2.3)shows that the generating function becomes simply(1−2 b,x + b 2)−µ−(d−1)/2=α∈N dbαR Bα(x),x∈R d.This is the generating function of one family of Appell’s biorthogonal polynomials and R Bα(x)is usually denoted by Vα(x)in the literature(see,for example,[8,Vol. II,Chapt12]or[7,Chapt2]).The deﬁnition of R Bαcomes from that of h-harmonics R(α,0)by the correspon-dence.If we consider Rβwithβ=(α,αd+1)and assume thatαd+1is an even inte-ger,then Rβleads to the orthogonal projection of the polynomial xα(1− x 2)αd+1/2 with respect to W Bκon B d.Furthermore,the correspondence also gives a generating function of these projections.2.3.Monomial orthogonal polynomials on the simplex.The h-spherical har-monics associated to(1.1)are also related to orthogonal polynomials associated tothe weight functions W Tκin(1.3).If Y∈H d+12n (h2κ)is an h-harmonic that is evenin each of its variables,then Y can be written as(2.9)Y(y)=r n P(x21,...,x2d),y=r(x1,...,x d,x d+1),r= y .The polynomial P(x),x=(x1,...,x d)∈R d is an orthogonal polynomial of degree n in d variables with respect to W Tκon T d.Moreover,this deﬁnes an one-to-one correspondence between the two sets of polynomials([18]).10YUAN XUSince the simplex T has a natural symmetry in terms of(x1,...,x d,x d+1), x d+1=1−|x|,we use the homogeneous coordinates X:=(x1,...,x d,x d+1).For the monomial h-harmonics deﬁned in Deﬁnition2.1,the polynomial R2αis even in each of its variables,which corresponds to,under(2.9),monomial orthogonal polynomials R Tαin V d n(W Tκ)in the homogeneous coordinates X.This leads to the following deﬁnitionDeﬁnition2.9.Letρ=|κ|+d−12.Deﬁne polynomials R Tα(x),α∈N d0,bycκ[−1,1]d+11(1−2(b1x1t1+...+b d+1x d+1t d+1)+ b 2)ρd+1i=1(1−t2i)κi−1dt=α∈N d+1b2α R Tα(x),x∈T d,x d+1=1−|x|.The main properties of R Tαare summarized in the following proposition. Proposition2.10.For eachα∈N d+1with|α|=n,the polynomialsR T α(x)=22|α|(ρ)2|α|(2α)!(1/2)α(κ+1/2)αR Tα(x),whereR Tα(x)=XαF B−α,−α−κ+12;−2|α|−ρ+1;1x1,...,1x d+1=(−1)n(κ+1)α(n+|κ|+d)nF A(|α|+|κ|+d,−α;κ+1;X)are orthogonal polynomials with respect to W Tκon the simplex T d.Moreover, R Tα(x)=Xα−Qα(x),where Qαis a polynomial of degree at most n−1,and {R Tα,α=(α ,0),|α|=n}is a basis for the subspace of orthogonal polynomials of degree n.Proof.We go back to the generating function of h-harmonics in Deﬁnition2.1. The explicit formula of Rαshows that Rα(x)is even in each of its variables only if eachαi is even for i=1,...,d+1.Letε∈{−1,1}d+1.Then Rα(xε)= Rα(ε1x1,...,εd+1x d+1)=εαRα(x).It follows thatβ∈N d+10b2βR2β(x)=12α∈N d+1bαε∈{−1,1}d+1Rα(xε).On the other hand,using the explicit formula of Vκ,the generating function gives1 2d+1ε∈{−1,1}d+1α∈N d+1bαRα(xε)=cκ[−1,1]d+1ε∈{−1,1}d+1d+1i=1(1+t i)(1−t2i)κi−1(1−2(b1x1t1ε1+...+b d+1x d+1t d+1εd+1)+ b 2)ρdtfor x =1.Changing variables t i→t iεi,the fact thatεd+1i=1(1+εi t i)=2d+1shows that the generating function of R2β(x)agrees with the generating function ofR Tβ(x21,...,x2d+1)in Deﬁnition2.9.Consequently,the formulae of R Tαfollow fromthe corresponding ones for R2α.The polynomial R Tαis homogeneous in ingMONOMIAL ORTHOGONAL POLYNOMIALS 11the correspondence (2.9)between orthogonal polynomials on S d and on T d ,we seethat R T αare orthogonal with respect to W T κ.If αd +1=0,then R Tα(x )=x α−Q α,which proves the last statement of the proposition.In the case of αd +1=0,the explicit formula of R T αshows that R T(α,0)(x )=x α−Q α(x );setting b d +1=0in Deﬁnition 2.9gives the generating function of R T (α,0).The explicit formula of R T (α,0)can be found in [7],which appeared earlierin the literature in some special cases.The generating function of R Tαappears to be new in all cases.We note that if all κi =0,then the integrals in the Deﬁnition 2.9disappear,so that the generating function in the case of the Chebyshev weight function W T (x )=(x 1...x d (1−|x |))−1/2is simply (1−2 b,x + b 2)−1.3.Symmetric monomial orthogonal polynomialsLet S d +1denote the symmetric group of d +1objects.For a permutation w ∈S d +1we write xw =(x w (1),...,x w (d +1))and deﬁne T (w )f (x )=f (xw ).If T (w )f =f for all w ∈S d +1,we say that f is invariant under S d +1.For α∈N d 0and w ∈S d +1,we deﬁne the action of w on αby (αw )i =αw −1(i ).Using this deﬁnition we have (xw )α=x αw .3.1.Symmetric monomial h -harmonics.In this section,we assume that κ1=...=κd +1=κ.Then the weight function h κin (1.1)is invariant under S d +1.LetH d +1n(h 2κ,S )denote the subspace of h -harmonics in H d +1n (h 2κ)invariant under the group S d +1.Our goal is to give an explicit basis for H d +1n (h 2κ,S ).A partition λof d +1parts is an element in N d +1such that λ1≥λ2≥...≥λd +1.Let Ωd +1denote the set of partitions of d +1parts.LetΩd +1n ={λ∈N d +10:λ1≥λ2≥...≥λd +1,|λ|=n },the set of d +1parts partitions of size n ,and let Λd +1n={λ∈Ωn :λ1=λ2}.For a partition λthe monomial symmetric polynomial m λis deﬁned by ([14])m λ(x )={x α:αbeing distinct permutations of λ}.Let B d +1denote the hyperoctadedral group,which is a semi-direct product of Z d +12and S d +1.A function f is invariant under B d +1if f (x )=g (x 21,...,x 2d +1)and g is invariant under S d +1.Since the weight function h κ(x )= d +1i =1|x i |κis invariant under B d +1,the monomials x αand x βare automatically orthogonal whenever αand βare of diﬀerent parity.Hence,closely related to H d +1n(h 2κ,S )is the space H d +1n (h 2κ,B ),the subspace of h -harmonics in H d +1n (h 2κ)invariant under B d +1.Re-cently Dunkl [6]gave an explicit basis for H d +1n (h 2κ,B )in the form of(3.1)p λ(x )=m λ(x 2)+ c µm µ(x 2):µ∈Ωd +1n ,µi ≤λi ,2≤i ≤d +1,µ=λ ,where x 2=(x 21,...,x 2d +1)and the coeﬃcients c µwere determined explicitly,andproved that the set {p λ:λ∈Λd +1n }is a basis of H d +1n (h 2κ,B ).Using the explicit formula of R αwe give a basis for H d +1n (h 2κ,S )in this section.Let S d +1(λ)denote the stabilizer of λ,S d +1(λ)={w ∈S d +1:λw =λ}.Then we can write m λ= x λw with the summation over all coset representatives of the subgroup S d +1(λ)of S d +1,which we denote by S d +1/S d +1(λ),it contains all w such that λi =λj and i <j implies w (i )<w (j ).12YUAN XUDeﬁnition3.1.Letλ∈Ωd+1.DeﬁneSλ(x)=w∈S d+1/S d+1(λ)Rλw(x).Proposition3.2.Forλ∈Ωd+1n ,the polynomial Sλ=proj n mλis an element ofH d+1n (h2κ;S).Moreover,the set{Sλ:λ∈Λd+1n}is a basis of H d+1n(h2κ,S).Proof.The deﬁnition of Sλand the fact that Rα(x)=xα+ x 2Qα(x)shows Sλ(x)=w∈S d+1/S d+1(λ)(xλw+ x 2Qλw(x))=mλ(x)+ x 2Q(x),where Q∈Πd+1n−1.Also Sλ∈H d+1n(h2κ)since each Rαdoes.Hence,Sλ=proj n mλ.It follows from the part(2)of Proposition2.5that,Sλ(x)=(−1)n2n(λ)nx 2ρ+2n mλ(D)( x −2ρ),which shows that Sλis symmetric.Since dim H d+1n (h2κ,S)=#Λd+1n,we see that{Sλ:λ∈Λd+1n }is a basis of H d+1n(h2κ;S).The fact that Sλis a symmetric polynomial also follows from a general statement about the best approximation by polynomials,proved in[1]for L p(S d)and the proof carries over to the case L p(S d;h2κ).Since the proof is short,we repeat it here.Letf p=c hS d|f(y)|p h2κ(y)dω(y)1/p,for1≤p<∞and let f ∞be the uniform norm on S d.Proposition3.3.If f is invariant under S d+1then the best approximation of f in the space L p(S d,h2κ)by polynomials of degree less than n is attained by symmetric polynomials.Proof.Let P∈Πd+1n−1.Sinceκ1=...=κd+1,hκis invariant under the symmetric group,and so is the norms of the space L p(S d;h2κ).Hence the triangle inequality and the fact that f is symmetric givesf−P p=1(d+1)!w∈S d+1f(xw)−P(xw) p≥1(d+1)!w∈S d+1f(xw)−w∈S d+1P(xw)p= f−P∗ p,where P∗is the symmetrization of P.Since P∗∈Πd+1n ,this shows that thebest approximation of f can be attained by symmetric polynomials of the same degree.The best approximation in L2(S d;h2κ)by polynomials is unique,so that a best ap-proximation polynomial to a symmetric function must be a symmetric polynomial. Thus,the above proposition applies to Sλ,as Sλ−mλis the best approximation to mλin L2(S d;h2κ)by polynomials of lower degrees.From the deﬁnition of Sλ,it is not immediately clear that Sλis symmetric.Next we give an explicit formula of Sλin terms of monomial symmetric functions and powers of x .We start with the following simple observation:。

机器证明-埃尔多斯-谢卡雷斯问题说明书

Mechanical Proving for ERDÖS-SZEKERES Problem1Meijing ShanInstitute of Information science and Technology,East China University of Political Science and Law, Shanghai, China. 201620Keywords: Erdös-Szekeres problem, Automated deduction, Mechanical provingAbstract：The Erdös-Szekeres problem was an open unsolved problem in computational geometry and related fields from 1935. Many results about it have been shown. The main concern of this paper is not only show how to prove this problem with automated deduction methods and tools but also contribute to the significance of automated theorem proving in mathematics using advanced computing technology. The present case is engaged in contributing to prove or disprove this conjecture and then solve this problem. The key advantage of our method is to utilize the mechanical proving instead of the traditional proof and this method could improve the arithmetic efficiency.IntroductionThe following famous problem has attracted more and more attention of many mathematicians [3, 6, 12, 16] due to its beauty and elementary character. Finding the exact value of N(n) turns out to be a very challenging problem. The problem is very easy to explain and understand.The Erdös-Szekeres Problem 1.1 [4, 15]. For any integer n ≥ 3, determine the smallest positive integer N(n) such that any set of at least N(n) points in generalposition in the plane contains n points that are the vertices of a convex n-gon.A set of points in the plane is said to be in the general position if it contains no three points on a line. This problem was also called Happy Ending Problem by Erdös, because two investigators Esther Klein and George Szekeres who first worked on the problem became engaged and subsequently married[8, 17].The interest of Erdös and Szekeres in this problem was initiated by Esther Klein(later Mrs. Szekeres), who observed that from 5 points of the plane of which no three lie on the same straight line it is always possible to select 4 points determining convex quadrilateral. There are three distinct types of five points in the plane, as shown in Figure 1.Figure 1. Three cases for 5 points.In any case of the Figure 1, one can find at least one convex quadrilateral determined by the points. Klein [4] suggested the following more general problem: Can we find for a given n a number N(n) such that from any set containing at least N points it is possible to select n points forming a convex polygon?As observed by Erdös and Szederes [4], there are two particular questions:(1) Does the number N corresponding to n exist?(2) If so, how is the least N(n) determined as a function of n?1T his work was financially supported by Humanity and Social Science Youth foundation of Ministry of Education of China (No. 14YJCZH020).They proved the existence of the number N(n) by two different methods. The first one is a combinatorial theorem of Ramsey. The foundation of the second one isbased on some geometrical and combinatorial considerations. And then they formulated the following conjecture.Conjecture 1.1 N (n ) = 2n −2 + 1 for all n ≥ 3.Despite its elementary characters and the efforts of many researchers, the ErdösSzekeres problem is solved for the value n = 3, 4 and 5 only. The case n = 3 istrivial, and n = 4 is due to Klein. The equality N (5) = 9 was proved by E. Makaiwhile the published proof by Kalbfleisch [11] and then Bonnice [2] and Lovasz [13]independently published the much simpler proofs. The bottle neck of this problem now is when n >5, how to prove or disprove the conjecture.About this problem, the best currently known bounds are(1)Where is a binomial coefficient. The lower bound was obtained by Erdös andSzekeres [4] and the upper bound is due to Töth and Valtr [18]. The lower bound issupposed to be sharp, according to conjecture 1.1.In this paper we use the automated deduction method and tools [1, 19-21] to establish a mechanical method for proving problem instead of the manual proof. We hope the method might give a rise to substantially promote this unsolved problem.The rest content of the paper is indicated by section headings as follows: Section 2 preliminaries, Section 3 main results, Section 4 conclusion and remarks.PreliminariesIn this Section, we present some algorithms that would help us develop our method in next section.Algorithm 2.1. Modified Cylindrical Algebraic Decomposition (CAD). Due to the problem statement, the proof should consider all kinds of points’ positions on the plane.The Cylindrical Algebraic Decomposition [5, 14] of R n adapted to a set ofpolynomials which is a partition of R d is cells (simple connected subsets of R d )such that each input polynomial has a constant sign on each cell. Basically, thealgorithm computes recursively at least one point in each cell (so that one can testthe cells that verify a fixed sign condition). The sample points Pi got by originalCAD are always more than one on each cell. We modify the procedure to have asample point on each cell of the final cell decomposition, by the rule that Pi has a constant sign on each cell. We elaborate the main idea under lying our method by showing how our main algorithm evolved from the original one.Here, we describe it as follows.Algorithm. MCADInput: A set F of polynomialsOutput: A F-sign-invariant CAD of R nStep 1. Projection. Compute the projection polynomials which using exclusivelyoperations, and receive some (n − 1) -variate polynomials.Step 2. Recur. Apply the algorithm recursively to compute a CAD of R n −1which Q(F )is sign-invariant.Step 3. Lifting. Lift the Q(F )-sign-invariant CAD of R n −1 up to a F-sign-invariant CAD of using the auxiliary polynomial Π(F ) of degree no largerthan d (F ) (d is the maximum degree of any polynomial in F).Step 4. Choice. Utilize the strategy that {Pi} has constant sign on each cell tochoose one sample point on each cell.Algorithm 2.2 (Graham Scan Algorithm) [9, 10, 22]We present one of the simplest algorithms used to find the convex hull fromsome points. Some basic definitions are provided in the field of Computational Geometry. This algorithm works in three phases:Input: A set S of pointsOutput: The convex hull of S.Step 1. Find an extreme point. The algorithm starts by picking a point in S known to be a vertex of the convex hull. This point is chosen to be with smallest y coordinate and guaranteed to be on the hull. If there are some points with the same smallest y coordinate, we will choose the point with largest x coordinate in them. In other words, we select the right most lowest point as the extreme point.Step 2. Sort the points. Having selected the base point which is called P0 , then the algorithm sorts the other points P in S by the increasing counter-clockwise (ccw) angle the line segment P0P makes with the x-axis. If there is a tie and two points have the same angle, discard the one that is closest to P0 .Step 3. Construct the convex hull. Build the hull by marching around the star shaped polygon, adding edges when we make a left turn, and back-tracking when we make a right turn. We end up with a star-shaped polygon, see Figure 3 (one in whichone special point, in this case the pivot, can “see” the whole polygon). Considering efficiency in Step 2, it is important to note that the comparison of sorting between two points P2 and P3 can be made without actually computing their angles. In fact, computing angles would use slow in accurate trigonometry functions, and doing these computations would be a bad mistake. Instead, one just observes that P2 would make a greater angle than P1 if (and only if) P2 lies on theleft side of the directed line segment P0P1 , see Figure 2.We make full use of this algorithm to judge whether the polygon received in every recursive step is a convex polygon or not. It is a decision method in our algorithm.To state the algorithm clearly, we will describe it in a style of pseudo-code.Algorithm. Graham Scan AlgorithmInput: A set S of points in the planeOutput: A list containing the vertex of the convex hullSelect the right most lowest point P0 in SFigure 3. Graham ScanFigure 2. Sort the points.Sort S angularly about P0 as a center.For ties, discard the closer points.Let S be the sorted array of points.Push S [1] = P0 and P1 onto a stack Ω.Let P1 = the top point on ΩLet P2 = the second top point on Ωwhile S [k ] P doif (S [k −1] is strictly left of the line S [k ] to S [k +1]), thenPush S [k −1] onto ΩelsePop the top point S [k ] off the stack Ω.fi;od;Main ResultsIn contrast with the traditional proof, this method presented following can show the convex polygons received in every step.The main idea of our algorithm: First, give randomly four points in general position in the plane, and then we use polynomials of points’ coordinates to represent the lines. We extend the 4-element set to some5-element sets. We do this by establishing the corresponding Modified Cylindrical Algebraic Decomposition (MCAD) and design an interactive program which allow the user to choose among the candidate one sample point in each cell. We use Graham Scan algorithm to determine whether or not there is a convex 5-gon (convexpentagon) in every set received. If some of the received 5-element sets have no convex 5-gon, then extend them to the 6-element sets by the strategy mentionedabove. Simultaneously, check whether or not each of the received 6-element sets has any convex hull at least with any convex 5-gon in the polygon. We implement theprogram repeatedly until can find a convex n-gon (n ≥ 5) in any set. We trace the processing of the extension and decision, and then draw a conclusion that N (5) = 9.To prove this approach, we write the following algorithm named “conv5”. Based on this algorithm can generate short and readable mechanical proving for the Erdös-Szekeres conjecture, including the case of n = 3, 4, 5. It consists of two main algorithms—Modified Cylindrical Algebraic Decomposition algorithm and Graham Scan algorithm and some sub-algorithms such as collinear, pol, sam, ponlist, min0,isleft, ord, conhull, point5, convex, G5,G6,Pmn.Algorithm Conv5Input: four points in the general position in the plane.Output: Any polygon with at least 9 points in general position in the planecontains a convex 5-gon. Step 1 [collinear]. Write the line polynomials with the given four points (basepoints).Step 2 [pol, sam, ponlist]. Illustrate how we utilize the CAD to find somesample points in the cell which built by the lines, and then with the base n points get n +1 -element sets.Step 3 [min0, isleft, conhull, point5, convex, G5]. Decide whether or not thereis a convex hull or a convex n-gon (n ≥ 5) in every set; if it is true, then stop; elsegoto Step 4.Step 4 [G6, Pmn]. Deal with the sets which have no convex n-gon (n ≥ 5).Recursively implement Step 2 and Step 3 process, until at least there is a convex 5-gon in any set. End Conv5.The key techniques of the algorithm are listed as follows:1. To reduce the complexity of the computation and increase the efficiency,when we check whether or not there is a convex 5-gon in the given points set,we utilize the strategy as follows:if there is a convex hull at least with 5 points in the points set thenpop this points setelif there is a convex 5-gon thenpop this points setelse “there is no convex 5-gon in this points set” go to next step2. Each convex n-gon (n ≥ 5) contains a convex 5-gon3. If there is no convex 4-gon, then there should be no convex 5-gon.Conclusion and RemarksBy the Maple procedure we have implemented the mechanical method for the conjecture in certain cases. Through observing the whole computational process, we obtain a certain answer that any set with at least 9 points in general position in the plane contains a convex 5-gon. This method can be generalized in an obvious way to arbitrary base points in the plane.For the mechanical method proposed here, on one hand it provided one of the promising direction for proving or disproving the Conjecture 1.1 (when n ≥ 6), even for handling with some unsolved problems in computational geometry. On the other hand, it gave one especially useful application of computer algebraic andautomated deduction. For further investigations, now we consider about the following problems:1. Does any set of at least 17 points in general position in the plane contains 6 points which are the vertices of a convex hexagon? Can we give the proof about N (6) existence and prove or disprovethe corresponding conclusion by mechanical proving? Now the best known conclusion about this is N (6) ≥ 27, if it exists.2. Erdös posed a similar problem on empty convex polygons. Whether or not we can give the automated proof to this problem?References[1] M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, omputational GeometryAlgorithm and Applications, (2nd ed.), Spring-Verlag, Berlin, Heideberg, New York, 1997.[2] W. E. Bonnice, On convex polygons determined by a finie planar set, Amer. Math. Monthly.[3] F. R. K. Chung and R. L. Graham, Forced Convex n-gons in the Plane, Discr. Comput. Geom. 19(1998), 367-371.[4] P. Erdös and G. Szekeres, A combinatiorial problem in geometry, Comositio Mathematica 2(1935), 463-470.[5] E. Collins George, Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition Lecture Notes In Computer Sciencevol. 33, Springer-Verlag, Berlin, pp. 134-183.[6] Kráolyi Gyula, An Erdös-Szekeres type problem in the plane.[7] X. R. Hou and Z. B. Zeng, An efficient Algorithm for Finding Sample Points of Algebraic Decomposition of Plane, Computer Application, 1997, (in Chinese).[8] P. Hofiman, The Man Who loved Only Numbers Hyperion, New York, 1998.[9] /ah/alganim/version0/Graham.html.[10] http://cgm.cs.mcgill.ca/ beezer/cs507/main.html.[11] J. D. Kalbfleisch, J. G. Kalbfleisch and R. G. Stanton, A combinatorial problem on convexregions, Proc. Louisiana Conf. Combinatorics, Graph Theory and Computing, Louisianna StateUniv., Baton Touge, La, Congr. Numer. 1 (1970), 180-188.[12] D. Kleitman and L. Pachter, Finding convex sets among points in the plane, Discr. Comput.Geom. 19, (1998), 405-410.[13] L. Lovasz, Combinatorial Problem and Exercises North-Holland, msterdam, 1979.[14] Bhubaneswar Mishra, Algorithmic Algebra, Springer-Verlag, 2001.[15] W. Morris and V. Soltan, The Erdös-Szekeres Problem on Points in ConvexPosition- A, Survey Bulletin of the American Mathematical Society, vol. 37.[16] L. Graham Ronald and Yao Frances, Finding the Convex Hull of a SimplePolygon Report No. STAN-CS-81-887, 1998.[17] B. Schechter, My Brain is Open Simon Schuster, New York, 1998.[18] G. Tóth and P. Valtr, Note on the Erdös-Szekeres theorem, Discr. Comput. Geom. 19 (1998), 457-459.[19] W. T. Wu, Basic Principles of Mechanical Theorem Proving in Geometries Science Press,Beijing, 1984, (Part on elementary geometries, in Chinese).[20] L. Yang and B. C. Xia, Automated Deduction in Real Geometry, Geometric Computation, WorldScientific, 2004.[21] L. Yang and Z. Z. Jing and X. R. Hou, Nonlinear Algebraic Equation System and AutomatedTheorem Proving, Shanghai Scientific and Technological Education Publishing House, Shanghai,1996, (in Chinese).[22] P. D. Zhou, Computational Geometry Design and Analysis in Chinese, TsingHua UniversityPress, 2005.。

第三届“丘成桐中学数学奖”获奖名单

第三届“丘成桐中学数学奖”获奖名单2010-12-20 10:37 来源：清华大学数学科学中心作者：清华大学数学科学中心[打印] [评论]丘成桐中学数学奖金奖（1名）上海市市北中学参赛队员：陈波宇指导老师：金荣生论文题目：Weierstrass函数在不可列的稠密集上不可导的一种证明丘成桐中学数学奖银奖（1名）中国人民大学附属中学参赛队员：刘頔、于伦指导老师：阳庆节论文题目：对凸形内部一类特殊点的研究丘成桐中学数学奖铜奖（3名）杭州第二中学参赛队员：干悦、陈宇戈、孙璐璐指导老师：金洁论文题目：从画正多边形的铰链到连杆轨迹广东广雅中学参赛队员：吴俊熹、熊奥林、刘哲指导老师：杨志明论文题目：瓦西列夫不等式的推广、加强与类似美国Milton Academy参赛队员：Farzan Vafa 指导老师：Yong Lin论文题目：A new definition of distance for graphs丘成桐中学数学奖优胜奖（5名）绍兴市第一中学参赛队员：胡煜中、陈璐萍、朱晖指导老师：凌晓锋论文题目：关于凸四边形最小个数问题绍兴市第一中学参赛队员：卢尔涵、林志成指导老师：凌晓锋论文题目：正整系数多项式的首位数字分布中国人民大学附属中学参赛队员：靳兆融、曾力玮指导老师：唐晓苗论文题目：三角形内切椭圆及其性质的研究美国Hazleton Area High School参赛队员：Keenan Monks 指导老师：Kenneth G. Monks论文题目：On supersingular elliptic curves and hypergeometric functions美国Liberty High School参赛队员：Benjamin Kraft 指导老师：Gregory Minton论文题目：Entries of random matrices丘成桐中学数学奖鼓励奖（8名）华东师范大学第二附属中学参赛队员：景琰杰、顾韬指导老师：施洪亮论文题目：高次剩余理论研究北师大实验中学参赛队员：施燕捷指导老师：黎栋材论文题目：汽车行驶最短时间武昌实验中学参赛队员：迟浩指导老师：瞿双清论文题目：关于京广铁路武汉枢纽列车最大通过能力的研究华中师范大学第一附属中学参赛队员：路康奕指导老师：王维佳论文题目：交通路口拥堵程度的量化与评价广东实验中学参赛队员：张舒瑶、张陈灵、陈明蕙指导老师：郭卫东论文题目：论Wilcoxon-Mann-Whitney检验的容许性华南师范大学附属中学参赛队员：霍泽恩指导老师：李维民论文题目：Heron 三角形的有理角平分线问题及与完全长方体的联系华南师范大学附属中学参赛队员：梁昊、张可天指导老师：黄毅文论文题目：金融投资中的复利期望模型深圳市外国语学校参赛队员：郭梦绮、袁可馨、萧桐桐指导老师：袁智斌论文题目：对选拔性考试量分法的实证性研究丘成桐中学应用数学科学奖金奖（1名）杭州外国语学校参赛队员：朱陶元敏、张允宜指导老师：潘俊论文题目：变速队伍中速度传递问题的研究及其应用丘成桐中学应用数学科学奖银奖（1名）新加坡NUS High School of Mathematics and Science参赛队员：Li Ang, Lim Sung Hyun, Wang Qi 指导老师：Chai Ming Huang 论文题目：Understanding flocking dynamics in nature丘成桐中学应用数学科学奖铜奖（2名）江门市第一中学参赛队员：张姝、李璟、李健斌指导老师：李凌山论文题目：Actuarial modeling on a children’s protection insur ance广东广雅中学参赛队员：黎骁旸、蔡宇指导老师：杨志明论文题目：墨西哥湾原油泄漏事件在推广Fay公式基础上的建模丘成桐中学应用数学科学奖鼓励奖（2名）郑州市第八中学参赛队员：汪晗、李晨、宋晨指导老师：刘正峰论文题目：二次拟合灰色预测模型在黄河水质发展趋势预测中的应用越南High School for the Gifted- Vietnam University of Hochiminh City 参赛队员：Nguyen Manh Tien, Tu Nguyen Thai Son 指导老师：Tran Nam Dung论文题目：A stone-picking game丘成桐中学数学奖组织奖（3名）中国人民大学附属中学绍兴市第一中学广东广雅中学第四届丘成桐中学数学奖获奖名单金奖学校：美国Sidwell Friends Upper School题目：Shock Pro Gas Dynamics in Thermal Nonequilibrium参赛队员：Wang Xie指导老师：Luo Tao银奖学校：深圳中学题目：存在一个非VAN DOUWEN极大几乎不相交族队员：何卓东指导老师：张承宇铜奖（共2项，排名不分先后）学校：北京十一学校题目：三次函数切割线的斜率关系参赛队员：曾文远指导老师：潘国双学校：安徽省合肥市168中学题目：裴蜀定理的加强证明参赛队员：丁宇堃指导老师：孙文海优胜奖（共5项，排名不分先后）学校：华东师范大学第二附属中学题目：Euler-Maclaurin公式的推广及其应用参赛队员：景琰杰指导老师：施洪亮学校：河南省实验中学题目：曲线的拆解参赛队员：汪晗刘婧孜宋晨指导老师：李新德学校：东北育才中学题目：经典趣题“老鼠与毒药问题”推广研究参赛队员：白天衣项思陶指导老师：张雷学校：辽宁省鞍山市第一中学题目：拓扑和的推广参赛队员：夏铭辰指导老师：张继红学校：广东实验中学题目：树的模染色数参赛队员：王家林源洁莹指导老师：郭卫东鼓励奖（共4项，排名不分先后）学校：浙江大学附属中学题目：数独的计数、分类与图案设计参赛队员：李梦鸽指导老师：李刚豪学校：江苏省苏州第十中学校题目：关于错排问题的思考与讨论参赛队员：钱炘祺于浩佳指导老师：吉剑锋学校：郑州第八中学题目：树枝的分形参赛队员：王佳瑞李明哲梁栋指导老师：刘正峰学校：郑州一高题目：基于防雷接地需求的土壤结构模型研究参赛队员：李豆豆指导老师：曹恒阁获得组织奖的学校（共2项，排名不分先后）杭州外国语学校广东省实验中学第五届丘成桐中学数学奖获奖名单金奖学校：清华大学附属中学题目：论两个函数方程解析解的渐近性质参赛队员：邵城阳指导老师：杨利军银奖学校：杭州第二中学题目：数学物理中的一个丢番图问题参赛队员：任之杨东辰指导老师：斯理炯铜奖（共3项，排名不分先后）学校：杭州外国语学校题目：一类离散最值问题的探究参赛队员：蔡煜晟赵海洲指导老师：张传鹏学校：广东实验中学题目：含Euler数和Bernounlli数的恒等式新探参赛队员：魏锐波储岸均指导老师：张俊杰学校：华南师范大学附属中学题目：莫比乌斯带分割的结构与拓扑性质参赛队员：樊润竹李想指导老师：罗碎海优胜奖（共3项，排名不分先后）学校：清华大学附属中学题目：扫雷游戏中数字和的最大值探究参赛队员：张益深指导老师：王殿军学校：北京十一学校题目：差分方程与微分方程间的关系及其解的性质的研究参赛队员：陈博轩指导老师：潘国双学校：华南师范大学附属中学题目：N线坐标体系及应用参赛队员：余欣航指导老师：李兴怀获得组织奖的学校（共3项，排名不分先后）华南师范大学附属中学清华大学附属中学杭州外国语学校第三届丘成桐中学应用数学科学奖获奖名单银奖学校：南京外国语学校题目：最优交通拥堵费定价研究队员：陈宗灿杜楠指导老师：龚强铜奖（共3项，排名不分先后）学校：南京外国语学校题目：一种工业上测量椭圆截面积和椭球体积的新方法参赛队员：尤宸超田汉指导老师：朱胜强学校：Weston High School, USA题目：Computationally Determining the Dimensions of the Homology Groups of Directed Graphs 参赛队员：Ariya R. Shajii指导老师：Gabor Lippner学校：清华大学附属中学题目：CG图和形独基本性质探究参赛队员：王芝菁指导老师：杨青明优胜奖（共5项，排名不分先后）学校：江苏省锡山高级中学题目：基于整数型延时差分方程组的基因表达调控的数学模型研究参赛队员：高竹指导老师：高军晖学校：清华大学附属中学题目：太阳时钟——计算时间的方法参赛队员：薛宇皓指导老师：周建军学校：澳门培正中学题目：云深不知处——代数学在云端储存上的应用参赛队员：谭知微指导老师：黄灿霖学校：福建省厦门第一中学题目：用微积分和微分方程解决家用热水器节能问题参赛队员：林梓楠谭天琪钱坤儿指导老师：徐小平学校：NUS High School, Singapore题目：Generalized Quantum Tic-Tac-Toe参赛队员：Ananya Kumar, ?Ang Yan Sheng指导老师：Chai Ming Huang金奖学校：St. Gregory College Preparatory School, USA题目：3D Surface Fabrication using Conformal Geometry参赛队员：Yuanqi Zhang指导老师：Xianfeng Gu银奖（共2项，排名不分先后）学校：北京市中关村中学题目：足球弧线球的数学分析方法参赛队员：杨祚儒富宏远计润达指导老师：潘凤易学校：NUS High School, Singapore题目：A General Algorithm of Flattening Convex Prismatoids参赛队员：Chenglei Li 、Jingqi Zhou指导老师：Minghuang Chai铜奖（共4项，排名不分先后）学校：北京市十一学校题目：一类Pell方程的可解性研究参赛队员：王嘉琦蔡立德指导老师：贾祥雪学校：South Brunswick High School, USA题目：q-Symmetric Polynomials and nilHecke Algebras参赛队员：Ritesh Ragavender指导老师：Alexander Ellis学校：吉林大学附属中学题目：Quantum Watermarking in M-band Wavelet Domain参赛队员：刘通徐旋指导老师：任玉莲学校：Arcadia High School, USA题目：Feature Identification for Colon Tumor Classification参赛队员：Anthony Hou指导老师：Ernie Esser优胜奖（共4项，排名不分先后）学校：澳门培正中学题目：A New Secure Distributed Storage Scheme for Cloud- Geometric and Algebraic design and Implementation参赛队员：张颢霆谭知微指导老师：黄灿霖学校：广东广雅中学题目：700阶以内有限群单性的探究参赛队员：邵芷茵赵子琪谢瑞恒指导老师：徐敏学校：清华大学附属中学题目：关于“青蛙跳几次，一米徘徊”概率问题的研究参赛队员：齐天博张胤泰指导老师：李劲松学校：杭州外国语学校题目：高层建筑安全疏散问题的研究及泛函背景下疏散方案设计的尝试参赛队员：周大桐王雨菡指导老师：潘俊鼓励奖（共8项，排名不分先后）学校：华东师范大学第二附属中学题目：形如ax+bx=cx的简单指数方程解的无理性判定参赛队员：钮敏学蔡偌箐指导老师：戴中元学校：南京外国语学校题目：基于基本体格检查提出综合肥胖指数的分析与应用参赛队员：崔珈铭王英之杰王子南指导老师：黄文龙学校：南京外国语学校题目：Leslie修正模型在南京市人口预测中的应用参赛队员：王伯文陈思齐楼嘉钰指导老师：王刚学校：华东师范大学第二附属中学题目：三角形中神奇的点参赛队员：潘星宇指导老师：戴中元学校：清华大学附属中学题目：地图投影变换在全球一张图量算中的应用参赛队员：马浩程李寅晓指导老师：周建军学校：福建省南安第一中学题目：Analysis and Comparison between the Algorithm Time Efficiency of Dijkstra and SPFA 参赛队员：谢新锋指导老师：林建源学校：广东实验中学题目：稳健的对数最优策略理论研究与实践参赛队员：郭屹峰黄辰光指导老师：刘江宁学校：广州市第六中学题目：“三国杀”中的数学问题分析参赛队员：杨卓潇李思聪伍思航指导老师：璩斌获得组织奖的学校（共4项，排名不分先后）广东广雅中学南京外国语学校华东师范大学第二附属中学清华大学附属中学。

Geekbench4CPUWorkloads

Geekbench 4 CPU WorkloadsIntroduction3 Platform Support4 Compilers4 Runtime5 Scores6 Cryptography Workloads7 AES7 Integer Workloads8 LZMA Compression8 JPEG Compression8 Canny8 Dijkstra9 SQLite9 LLVM9 HTML 5 Parse9 HTML 5 DOM9 PDF10 Lua10 Camera10 Floating Point Workloads11 GEMM11 FFT11 Gaussian Blur11 Ray Trace12 N-Body Physics12 Rigid Body Physics13 Speech Recognition13 Memory Workloads14 Memory Copy14 Memory Latency14 Memory Bandwidth 14IntroductionThis document outlines the CPU workloads included in Geekbench 4.Platform SupportPlatform Minimum Version Comment Android 5.0iOS9.0Linux14.04 LTSOS X10.11Windows7 CompilersPlatform Compiler Comment Android Clang 3.8NDK r12 iOS Xcode 7.3Linux Clang 3.8OS X Xcode 7.3Windows Visual Studio 2015RuntimeGeekbench 4 groups CPU workloads into two sections:1.Single-Core Workloads2.Multi-Core WorkloadsEach section is grouped into four subsections:1.Cryptography Workloads2.Integer Workloads3.Floating-Point Workloads4.Memory WorkloadsGeekbench inserts a pause (or gap) between each workload to minimize the effect thermal issues have on workload performance. Without this gap, workload that appear later in the benchmark would have lower scores than workload that appear earlier in the benchmark. The default gap is 2 seconds for both single-core and multi-core workloads.ScoresGeekbench 4 provides two composite scores: single-core and multi-core. These scores are computed using a weighted arithmetic mean of the subsection scores. The subsection scores are computed using the geometric mean of the scores of the workloads contained in that section.Subsection WeightCryptography5% Integer45% Floating Point30% Memory20%Cryptography WorkloadsAESThe Advanced Encryption Standard (AES) deﬁnes a symmetric block encryptionalgorithm. AES encryption is widely used to secure communication channels (e.g.,HTTPS) and to secure information (e.g., storage encryption, device encryption).The AES workload in Geekbench 4 encrypts a 32MB string using AES running in CTR mode with a 256-bit key. Geekbench will use AES instructions when available, and fall back to software implementations otherwise.Devices with superior AES performance can have a noticeable effect on the usability of the device. See, e.g., /gadgets/2015/03/review-the-new-moto-e-is-the-most-phone-you-can-get-for-150/3/Integer WorkloadsLZMA CompressionLZMA (Lempel-Ziv-Markov chain algorithm) is a lossless compression algorithm. Thealgorithm uses a dictionary compression scheme (the dictionary size is variable and can be as large as 4GB). LZMA features a high compression ratio (higher than bzip2).The LZMA workload compresses and decompresses an 450KB HTML ebook using the LZMA compression algorithm. The workload uses the LZMA SDK for the implementation of the core LZMA algorithm.This workload replaces the BZip2 Compress and BZip2 Decompress workloads fromGeekbench 3.JPEG CompressionJPEG is a lossy image compression algorithm. The algorithm works by encoding animage in 8x8 blocks. Each block is transformed using a discrete cosine transform (DCT).Each DCT coefﬁcient is quantized to achieve compression. The coarseness of thequantization determines the quality of the compressed image.The JPEG workload compresses and decompresses an image using the JPEGcompression method. The JPEG workload sets the quality parameter to “90”, acommonly-used setting for users who desire high-quality images.The workload uses libjpeg for the implementation of the core JPEG algorithm.This workload replaces the JPEG Compress and JPEG Decompress workloads fromGeekbench 3.CannyCanny Edge Detection is a widely used technique in image processing and computervision -- the paper that introduced the algorithm has over 20,000 citation according toGoogle Scholar. Canny is a component of many computer vision and image processing algorithms including registration, stereo correspondence, and object recognition.Canny is a sophisticated workload with four separate components:* Noise removal step (Gaussian Blur)* Gradient calculation* Minimum maximum threshold pass* Edge following step (BFS)This workload replaces the Sobel workload from Geekbench 3.DijkstraThe Dijkstra workload computes driving directions between a sequence of destinations.Similar techniques are used to compute paths in games, to route computer network trafﬁc, and to route driving directions. The data for this workload is based on Open Street Map data for the Waterloo Region. The data 79392 nodes and 162644 edges with integerweights approximating travel time along the section of road represented by the edge. The route includes a source and 9 ordered destinations.This workload will be similar to the Dijkstra workload from Geekbench 3.SQLiteSQLite is a self-contained SQL database engine, and is the most widely deployeddatabase engine in the world.The SQLite workload executes SQL queries against an in-memory database.Thedatabase is synthetically created to mimic ﬁnancial data, and is generated usingtechniques outline in “Quickly Generating Billion-Record Synthetic Databases” by J.Gray et al. The workload is designed to stress the underlying engine using a variety of SQLfeatures (such as primary and foreign keys) and query keywords such as: SELECT,COUNT, SUM, WHERE, GROUP BY, JOIN, INSERT, DISTINCT, and ORDER BY. This workload measures the transaction rate a device can sustain with an in-memory SQLdatabase.LLVMLLVM is a compiler infrastructure library that provides a source- and target-independent optimizer. LLVM also provides code-generation support for several architectures.The LLVM workload processes an LLVM IR (intermediate representation) ﬁle through the LLVM optimizer and code-generation routines. The LLVM IR ﬁle was generated from a 3,900 line C source ﬁle using Clang. The workload uses ARM as the target architecture for code generation.HTML 5 ParseConstruct a parse tree from a large HTML5 document using Gumbo, Google’s HTML5parsing library. The HTML5 document selected for this workload is the Wikipedia article “List of city nicknames in the United States”, a 1.2 MB document selected because itcontains the most internal references of any Wikipedia article and therefore contains alarge amount of links, list items, superscript numbers, and text. This workload tests how quickly a device can process an HTML5 document into a data structure required by web browsers.HTML 5 DOMThis workload constructs a HTML5 Document Object Model (DOM) based on thespeciﬁcation supplied at https:/// and performs manipulations of that DOM. The document selected for this workload is the ‘Processor Benchmarks’ page from . This workload performs several manipulations, including node insertion and deletion, and selective node class decoration, to measure how responsive those operations would be during a normal web browsing session.PDFThe Portable Document Format (PDF) is a standard ﬁle format used to present andexchange documents independent of software or hardware. PDF ﬁles are used innumerous ways, from government documents and forms to e-books.The PDF workload parses and renders a 29-page PDF document. The document ismostly text with a few small images placed throughout the document. The PDF workload uses the PDFium library (which is used by Google Chrome to display PDFs).LuaLua is a scripting language commonly used as an embedded scripting language in larger applications (notable examples include Adobe Photoshop Lightroom and World ofWarcraft).The Lua workload executes a Lua script using the standard Lua interpreter. The scriptparses data from a JSON ﬁle and uses Mustache to combine the data with a template to produce an HTML ﬁle.CameraCamera replicates a photo sharing application like Instagram. Camera merges several steps into one workload*AES decryption*AES key generation using PBKDF HMAC SHA1*SHA2 checksum generation*JSON parsing*JPEG compression and decompression*PNG decompression*Image compositing*Image ﬁlters (gaussian blur, contrast)*SQLite (INSERT the processed image metadata into a database)All steps run on the CPU and are not accelerated by the GPU.Floating Point WorkloadsGEMMGEMM (General Matrix Multiplication) computes the result C = AB + C, where A, B, and C are matrices.The GEMM workload uses 512x512 single-precision matrices. The GEMMimplementations are written using hand-tuned vector instructions for the followinginstruction sets:•AVX + FMA•AVX•SSE3•SSE2•ARMv8 NEON•ARMv7 NEONFFTFFT (Fast Fourier Transform) decomposes an input signal into a linear combination of a basis of trigonometric polynomials. FFT is a core algorithm in many signal-processingapplications.The FFT workload executes an FFT on an 32MB input buffer operating in 16KB chunks.This is similar to how FFT is used to perform frequency analysis in an audio processing application.The FFT implementations are written using hand-tuned vector instructions for the following instruction sets:•AVX2•AVX•SSE3•SSE2•ARMv8 NEON•ARMv7 NEONGaussian BlurGaussian Blur blurs an image using a Gaussian spatial ﬁlter. Gaussian Blurs are widely used in software, both in operating systems to provide interface effects, and in imageediting software to reduce detail and noise in an image. Gaussian Blurs are also used in computer vision applications to enhance image structures at different scales.The Gaussian Blur workload blurs an image using a Gaussian spatial ﬁlter. While theGaussian Blur implementation supports an arbitrary sigma, the workload uses a ﬁxedsigma of 1.0f. This sigma translate into a ﬁlter diameter of 9 pixels.Gaussian Blur replaces the Sharpen Image and Blur Image workloads from Geekbench 3. Ray TraceThe Ray Trace workload is an expanded implementation of the same workload fromGeekbench 3. Reﬂection, refraction, and translucency effects have been added, alongwith texture mapping and bump mapping. Ray Trace also implements soft shadows via area lights.The Ray Trace workload renders our version of the Cornell Box, which is traditionally used to test rendering software. The scene includes a reﬂective cylinder and a texture-mapped sphere. The two objects are surrounded by a mesh (the box) and a diffuse light source is positioned at the top of the box. The rendered image is 300x300 pixels.N-Body PhysicsComputes a 3D gravitation simulation using the Barnes-Hut method. To compute the exact gravitational force acting on a particular body x in a ﬁeld of N bodies requires N − 1 force computations. The Barnes-Hut method reduces the number of force computations byapproximating as a single body any tight cluster of bodies that is far away from x. It does this efﬁciently by dividing the space into octants — eight cubes of equal size — andrecursively subdividing each octant into octants, forming a tree, until each leaf octantcontains exactly one body. This recursive subdivision of the space requires ﬂoating point operations and non-contiguous memory accesses.Rigid Body PhysicsComputes a 2D physics simulation for rigid bodies that includes collisions and friction.The simulation is implemented using the popular Box2D physics library.Speech RecognitionPerform recognition of arbitrary English speech using PocketSphinx, a widely used library that uses HMM (Hidden Markov Models) to perform speech recognition. Using speech to interact with smartphones is becoming more popular with the introduction of Siri, Google Now, and Cortana, and this workload tests how quickly a device can process sound and recognize the words are being spoken.Memory WorkloadsMemory CopyMeasures the performance of the system-provided memcpy() routine using different sizes and randomized offsets in a large memory buffer. The sizes are selected based on the distribution of memcpy() calls in user applications.Memory LatencyMeasures the latency of system memory by traversing a circular linked-list. The nodes of the linked list are arranged in such a way to reduce the number of TLB cache misses (1 per page rather than 1 per access).Memory BandwidthMeasures sustained memory bandwidth. The implementation is written using hand-tuned vector instructions for the following instruction sets:•AVX•SSE2•ARMv7 NEONMemory Bandwidth is similar to the STREAM Scale workload from Geekbench 3.Measures sustained memory bandwidth. We will provide hand-vectorizedimplementations for Intel and ARM architectures.Similar to the STREAM workloads in Geekbench 3.。

PolesandZerosofTransferFunctions：传递函数的零点和极点

Poles and Zeros of Transfer FunctionsMost circuits we consider, in practice, are discrete (circuit parameters such as R s, L s and C s, appear in discrete lumps, rather than smears, as they do intransmission lines), time invariant (the values of circuit parameters such as R s, L s and C s, so not change with time), and linear (the i-v relationships of the circuit elements, such as R s, L s and C s, are given by mathematically linear relations of i , v and their derivatives). For such ordinary circuits, all transfer functions can be written as the ratio of polynomials in the variable, s :()()()N s T s D s = where ()N s is the numerator polynomial and ()D s is the denominator polynomial.A function such as ()T s which can be expressed as a ratio of polynomials iscalled a rational function . A polynomial in s , recall, is a weighted sum of the powers of s up to some finite maximum power of s . For example,()212102P s s s =++is a polynomial. Because its highest power of s is 2, this particular polynomial is said to be a second-degree polynomial. (The function s e -, as an example, is not a polynomial because its Taylor series expansion is an infinite series of terms of increasingly higher powers of s .)The degree of the denominator polynomial, ()D s , for a transfer function basically is given by the number of energy storage elements (inductors and capacitors) in the circuit, although series or parallel combinations of inductors or capacitors that can be reduced to a single element count only once. The degree of the numerator polynomial, ()N s , is usually less than or equal to the degree of the denominator polynomial.Any polynomial of degree n can be factored into the product of n terms of the form ()s a +, where a is a constant. For example, we can factor thepolynomial ()212102P s s s =++ as follows:()()()212102223P s s s s s =++=++In this example, the constants in each factor are real integers. This circumstance is atypical. Often, the constants are complex irrational numbers. For example, the polynomial ()21P s s s =-+ requires complex constants in its factors:()211122P s s s s s ⎛=-+=-+-- ⎝⎭⎝⎭ If ()N s is a polynomial of degree m and if ()D s is a polynomial of degree n , then we can write transfer functions in factored form:()()()()()()()1212m n s z s z s z T s K s p s p s p --⋅⋅⋅-=--⋅⋅⋅- where the constants 12,,...,m z z z are called the zeros of the transfer function, ()T s , the constants 12,,...,n p p p are called the poles of the transfer function and K is some constant. The poles and zeros are not necessarily unique. That is,some factors may be repeated. Note that ()T s is completely specified by its poles, its zeros and the constant K . That is, ()T s can be reconstructed exactly if we know these 1m n ++ quantities.Should the value of s ever equal any one of the zeros, 12,,...,m z z z , say k z , of ()T s , note that()0k T z =Hence, the name zeros. Of course some of the zeros may be complex, so for these, ()T s achieves zero value only when s takes on complex values. We thus will find it necessary to consider s to be a complex variable. That is, it can take on values in the complex s -plane, a plane with real and imaginary axes. In thiscomplex s-plane, for example, we can plot the zeros112z j =+ 212z=-that we found earlier for the polynomial ()211122P s s s s s ⎛=-+=-+-- ⎝⎭⎝⎭as follows:In this example, note that the two zeros of ()P s lie in symmetric positions with respect to the horizontal axis. In the language of complex numbers, z 1 and z 2 are said to be complex conjugates : their real parts are identical and their imaginary parts have the same magnitude but opposite signs. If two numbers, z 1 and z 2, are complex conjugates, the customary notation is that 21*z z = or, of course, 12*z z =. If we think of conjugation as an operator, it has the effect of reversing the sign ofthe imaginary part of a complex number. For example, if112z j =+and 212z =-then12*11*2222z j j z ⎛=+=- ⎝⎭We now demonstrate an important result known as the conjugate law for transfer functions : the poles and zeros of a transfer function,()()()N s T s D s = either (1) lie on the real axis of the s -plane or (2) occur in complex conjugate pairs.To begin demonstration of this result, note that the coefficients of s in ()N s and ()D s are real numbers because they are just combinations of the various (real) circuit parameters (R s, L s, Cs and so forth). For any polynomial, ()P s , with real coefficients for the powers of s , it is true that()()**P s P s = For a specific example, suppose()246116P s s s =++in which, we note, all coefficients of s are real. Thus, we see()()()2***46116P s s s =++ ()()**246116P s s s =++()()()()***P s P s P s =≡ The key to obtaining this result is that the coefficients of s in the polynomial are real.We now apply this result separately to the polynomials ()N s and ()D s . For any of the zeros of the transfer function, say, k z , we have()0k N z =If we take the complex conjugate of this equation, we see()*0k N z =From the conjugate law, we have()*0k N z = Thus, we have the result that if k z is a zero of ()T s , then *k z also is a zero of()T s . Thus, unless k z is real, it is a member of a complex conjugate pair of zeros. We conclude that the zeros of ()T s either are real or occur in complex conjugate pairs.What about the poles of ()T s ? For any of the poles, say, k p , we have()0k D p =If we take the complex conjugate of this equation, we see()*0k D p =From the conjugate law, we have()*0k D p =Thus, we have the result that if k p is a pole of ()T s , then *k p also is a pole of ()T s . Thus, unless k p is real, it is a member of a complex conjugate pair of poles.We conclude that the poles of ()T s either are real or occur in complex conjugate pairs.These two results, for the poles and zeros of ()T s , demonstrate the conjugate law for ()T s . Because of the conjugate law, a plot in the s -plane of the locations of the poles and zeros of ()T s are always symmetric about the real axis in the s -plane. Consider, for example,Because a pole-zero plot such as this shows the location of all poles and zeros of ()T s and because ()T s can be constructed from its poles and zeros (plus the constant, K , which can be determined if the value of ()T s is known for a single value of s ), the pole-zero plot can serve as an alternative representation to the rational (ratio of polynomials) function representation of ()T s .We have already mentioned that inverting ()out V s to obtain ()out v t can involve a lot of effort. This difficulty provides strong motivation to learn as much as possible about the behavior of ()out v t from investigating ()out V s without going through the trouble of inverting it. After all, ()out V s contains precisely the same information as ()out v t since ()out v t can be reconstructed from ()out V s . Laplace transform theoryprovides two results, the initial value and the final value theorems, that give bits of information about ()out v t directly from ()out V s : ()()0lim out out s v sV s +→∞= ()()0lim out out s v sV s +→∞=These results, valid when the limits exist, give the initial and final values of ()out v t directly from ()out V s without inverting ()out V s . The final value theorem can give information about the ultimate stability of an amplifier – if ()out v ∞→∞, then the amplifier clearly is unstable. Unfortunately, the output of unstable amplifiers typically is an exponentially growing sinusoid, which oscillates between ±∞. In such cases, the limit in the final value theorem does not exist, so that theorem provides no information. We obtain a more useful test for stability by applying a partial fraction expansion to ()out V s .For present purposes, we consider an amplifier to be stable if, in response to an impulse, ()t δ, to its input, the output, ()out v t , of the amplifier eventuallyapproaches zero. By applying an impulse to the input, we are applying all possible frequencies simultaneously. We will therefore excite any instability in the amplifier, regardless of its frequency. If the input is an impulse at 0t =, then we have()(){}(){}1in in V s L v t L t δ===where {}L ⋅ indicates Laplace transformation. Thus, for the purpose of examining stability of an amplifier, we have:()()()()()()()()()()1212m out in n s z s z s z V s T s V s T s K s p s p s p --⋅⋅⋅-===--⋅⋅⋅- The idea of a partial fraction expansion is that a rational function, such as ()T s , can be expanded as the sum of simpler rational functions, each of which has only one pole (although, in certain cases the pole can be repeated). If, for simplicity, we assume that none of the poles of ()T s is repeated, its partial fraction expansion takes the form()1212......k n k nA A A A T s s p s p s p s p ≡+++++---- where ,1,2,...,k A k n =, are complex constants that are chosen so that the identity holds for all s . If ()T s has repeated poles, then its partial fractionexpansion is slightly more complicated, but our main results do not change. As we’ve seen, each of the poles 12,,...,n p p p can have both real and imaginaryparts: k k k p j οω=+Thus, we can write()121122......k n out k k n nA A A A V s s j s j s j s j οωοωοωοω=+++++-------- By taking the inverse Laplace transform (from a table of transforms), we see()112212......k k n n t j t t j t t j t t j t out k n v t Ae A e A e A e οωοωοωοω++++=+++++ Because the poles of ()T s are either real or occur in complex conjugate pairs, more detailed calculations show that the terms can be grouped to show explicitly that the right hand side of this equation is real, as required (since ()out v t must be real). Those details are of little concern to our stability analysis, however. We note that if all of the 0,1,2,...,k k n ο<=, then each term produces damped oscillations, so that the output, ()out v t , approaches zero for long times. That is, an amplifier is stable if all poles of its transfer function, ()T s , lie in the left half of the s-plane . If even one 0k ο≥, however, then the corresponding term produces exponentially growing oscillations for long times. That is , an amplifier is unstable if at least one pole of its transfer function lies in the right half s-plane . Incidentally, we consider a transfer function with poles on the imaginary axis to be unstable, even though the corresponding terms do not approach infinity because ()out v t does not approach zero after long times. It oscillates.。

高等代数中的英文单词和短语

高等代数
Higher Algebra
第一章
Chapter 1 Polynomials
实系数多项式
real coefficient polynomial
有理系数多项式
rational coefficient polynomial
复系数多项式
complex coefficient polynomial
函数
单根
single root
重根
multiple root
第二章
Chapter 2 Determinants
排列
permutation
对换
transposition
行列式
determinant
余子式
cofactors term
克拉默法则
Cramer rule
拉普拉斯定理
Laplacetheorem
第三章
矩阵的行秩
row rank of matrix
左分配率
left distributive law
右分配率
right distributive law
结合律
associative law
第五章
Chapter 5Quadratic Forms
二次型
quadratic form
对称矩阵
symmetric matrix
可对角化矩阵
diagonalizable matrix
线性替换
linear substitution
标准型
normal form
正定二次型
positive definite quadratic form
正定矩阵
positive definite matrix

数学词汇

mapimagefixed pointcomposite functionone to one / injectiveonto / surjectivebijectiveinverse functionreciprocaldenominatorsymmetric with respect to the y axis / the origin / the line y=x abscissax/y interceptordinateanalytic geometryparabolahyperbolaconic sections 二次曲线系coefficientspoint-slope formuladirectrix 准线vertex focuslatus rectum 过焦点平行于准线的弦radius, centertangent linedegenerate 退化major axis / minor axiseccentricity e=c/adifference 差branch 支asymptotes 渐近线focal axispolynomial equationsquadratic polynomial 二次多项式quadratic formuladiscriminant判别式division algorithm / remainder theoremquotientfundamental theorem of algebra multiplicity 重根conjugate radical 共轭根the complex conjugate 共轭复数‘monic (an=1)the rational roots theoremlogarithm 对数GRE用log x表示lnx trigonometry 三角几何complementary 互余cos = complementary sinetangent / cotangent / secant / cosecant terminal side 终边quadrant 象限arbitrary angle 钝角trig function 三角函数periodicity 周期性periodequidistant 等距sequence 序列convergent / divergent(minus) infinitymonotonic 单调bounded 有界the sandwich (or squeeze) theorem approach A from above (右逼近A) continuous functionThe Intermediate Value Theorem 中值定理derivative 导数secant line 割线tangent line 切线normal line 法线differential 微分的linear appropriationimplicit differentiation 隐函数求导concave up (convex) f’’>0concave down (concave) f’’<0inflection point 拐点local minimum / absolute minimumcritical point / stationary point (f’=0) nth-derivative test fn>0 极小，反之极大adjacent sides 毗连的边relate rates dy/dt= r dx/dt indefinite integration 不定积分intersect 曲线相交rectangular (or Cartesian) coordinates polar coordinatescardioids 心形线r=2a(1+cos(sita)) solids of revolution 旋转体infinite series 无穷极数harmonic seriesp-seriesalternation seriespower seriesthe radius of convergencethe interval of convergence arccosine functionarcsine functiondomainadjoint 伴随阵determinantexpected valueprobability density function derivativeinflection pointrankeigenvalueeigenvetoreigenspacesubsetpolyhedron 多面体vertices / vertexinverse of the matrixorthogonal 正交height 多项式系数绝对值和+最高次tracepolynomialidempotent 幂等A2=Anilpotent 幂零scalar 数量阵fixed pointthe qth roots of unitycoset 陪集dot/scalar product 点积projAB B在A的投影cross producttriple scalar product （A*B）•C magnitude 模parametric equation 参数方程symmetric equation （直线）对称式generator, elements cracking p111 arbitrarylevel curve of height 等高线contour curve 轮廓线（被平面截的截面）hyperboloid 双曲面circular poraboloidcylindrical coordinatesspherical coordinatespartial derivativedirectional derivativesgradientsaddle point cracking p131Hessian matrixline integralThe Fundamental Theorem of Calculus for Line Integral 势场内线积分只与起止点有关gradient field My=Nxconservative 值与路值无关Green Theorem cracking p152ordinary/partial differential equation (ODE/PDE)homogeneous of degree n n阶齐次exact differentialintegrating function 积分因子inconsistent （线性方程组）无解commutative 交换的invertible 可逆的associative 传递性coefficient matrixaugmented matrix 增广矩阵Gaussian Eliminationechelon formparameter 参数nullspacelinear combinationspan 几个向量的所有线性组合trivial combination即linearly independentbasis a minimal spanning set for a vector space dimension 基中向量数normal vector 法向量column space / row spaceLaplace expansions 即按某行/列展开adjugate matrix 共轭矩阵Cramer’s Rule克莱莫法则scalar 数乘linear operation=linear transformationkernel / nullity / range / rankRank plus Nullity TheoremCayley-Hamilton Theorem p(A)=0 divisibility, factor, multipleprime number, compositegreatest common divisor (gcd)least common multiple (lcm)the congruence equation ax=b(mod n)the Euclidean Algorithm 欧氏算法cracking p222 congruence 余数binary operation on S S S*Sassociate ：a•(b•c)=(a•b)•c semigroup条件identity 单位元semigroup+identity=monoidmonoid+inverse=groupabelian groupgeneral/special linear groupSn symmetric group对称群（阶为n!）S3为最小的6阶非阿贝尔对称群alternating group 置换群（同上）polygon 多边形equilateral triangle 等边三角形isosceles triangle 等腰三角形Dn nth dihedral group :order(Dn)=2n additive group of integers modulo n multiplicative group of integers modulo p cyclic groupKlein four-group, or viergruppeproper subgroupnontrivial subgroupgenerators 生成元finitely generatedisomorphism 同构homomorphism 同态monomorphism 单同态epimorphism 满同态endomorphism 自同态automorphism 自同构direct product (a, b) cracking p237direct sum 同上if abelianelementary divisors/ invariant factors cracking p238normal subgruoup 正规子群inner automorphism induced by aunity 环乘法单位元unit 存在乘法逆的非零元素ring with unity 幺环commutative ringsubringcharacteristicring of integersring of integers modulo n (Zn, +, •)ring of Gaussian integers Z(i)ring of polynomials in x over R R[x]ring of real-valued functions on R RR 交换幺环evaluation (or substitution) homomorphism at a cracking p249 Frobenius endomorphism f(a)=app is a prime numberbinomial theorem 二项式定理integral domain 整环left/right zero divisor 零因子cancellation law a!=0, ac=ab, them c=bdivision ring 无零因子的环field= commutative division ring又，有限整环是域strictly-skew field= noncommutative division ring体real quatenion 四元素体Boolean ring 该环中元素idempotent subset > supersetuniversal setcomplement of B relative to A A-B union / intersectionsymmetric difference (A-B)U(B-A)Cartesian product 笛卡尔积open / closed intervalcardinality (cardinal number) 元素数countably infinitealgebraic numbers cracking p267 power set of Alevels of infinitycardinal number of continuumtranscendental numberscombination, permutationbinomial coefficientpigeonhole principle 抽屉原理probabilityBoolean algebra (or algebra) of sets on S: E指the power set of S 的子集probability measure on E cracking p274distribution functionvariance, standard deviationthe normal distribution 正态分布standard normal distributionbinomial distribution 二项分布imaginary unit iprinciple argument 幅角主值sample space (S), outcomes (S中元素), events（E中元素，S的子集）independent独立, mutually exclusive相斥Bernoulli trialspolar form, exponential formprincipal logarithmprincipal value of zwhyperbolic function 双曲函数Laplace equation / harmonic uxx+uyy=0entire function 在复平面内解析disk of convergencepunctured open disk cracking p312singularity, isolated singularitypole of order nsimple pole (n=1) double pole (n=2)essential singularityannulus 环面singular (or principal ) part / analytic partresidueHausdorff spaceindiscrete / trivial topologyinterior, exterior, boundary, limit point, closure interior+boundary=closurelower-limit topology B=[a,b)connectedcovering, open coveringcompactnessnorm of a point cracking p290 Euclidean metric 欧氏度量square metricopen map != continuous 一来一去，方向反homeomorphism = continuous + open map upper bound, bounded abovelub=suremum (sup)glb=infimum (inf)complete space = no holesLebesgue measurable setssingletonLebesgue measurable functionLebsegue integrablestep function cracking p296index of a group 子群中元素的最小公共阶tangent 相切loop 循环inradius 内径trajectory 轨迹centroid 重心clusterpartial fraction expansionsufficient / necessary condition先是issue，45分钟，完了以后紧接着argue，然后是10分钟的休息时间，你可以选择跳过。

关于一个多项式序列的对称关系

第38卷第4期西南师范大学学报(自然科学版)2013年4月V o l .38 N o .4 J o u r n a l o f S o u t h w e s t C h i n aN o r m a lU n i v e r s i t y (N a t u r a l S c i e n c eE d i t i o n )A pr .2013文章编号:10005471(2013)04002803关于一个多项式序列的对称关系①何圆1,2, 张文鹏21.昆明理工大学理学院,昆明650500;2.西北大学数学系,西安710127摘要:研究了一个与广义B e r n o u l l i 多项式㊁高阶A p o s t o l -B e r n o n l l i 多项式和高阶A p o s t o l -E u l e r 多项式相关的多项式序列.运用组合技巧建立了该多项式序列与幂和之间的两个对称关系.关键词:广义B e r n o u l l i 多项式;A p o s t o l -B e r n o u l l i 多项式;A p o s t o l -E u l e r 多项式;幂和中图分类号:O 156.4文献标志码:A令f 是一个正整数,g 是定义在整数集上的周期为f 的一个函数.文献[1]介绍了如下广义B e r n o u l l i 多项式B n ,g (x ):ðf-1i =0g (i )t e (i +x )t e f t -1=ðɕn =0B n ,g (x )t nn !(1)特别地,f =1和g (0)=1的情形分别给出了B e r n o u l l i 多项式B n (x )和B e r n o u l l i 数B n =B n (0).下面是文献[2]给出的一个关于B e r n o u l l i 数的循环关系:B n =1a (1-a n )ðn -1i =0n æèçöø÷i a i B i ða -1j =1j n -i n ȡ1,a ȡ2(2)作为应用,文献[3]给出了著名的S t a u d t -C l a u s e n 定理㊁C a r l i t z 定理㊁F r o b e n i u s 定理和R a m a n u j a n 定理的简单证明.鉴于式(2)的重要性,文献[4-6]对其作了进一步的研究.源于以上工作,本文研究了一个多项式序列{g(r )n ,g ,λ(x )}ɕn =0:ðf-1i =0g (i )λi e i tg λ(t ())r e x t=ðɕn =0g (r)n ,g ,λ(x )t nn !(3)其中r 是非负整数,f 是正整数,g 是定义在整数集上的周期为f 的一个函数,λ是一个变量,gλ(t )是带有变量λ的一个幂级数.通过运用一些组合技巧,我们建立了该多项式序列{g(r)n ,g ,λ(x )}与幂和S n ,g ,λ(k )=g (0)λ00n +g (1)λ11n + +g (k -1)λk -1(k -1)n 之间的两个对称关系.作为应用,文献[5-6]中相关的结果被作为特殊情况得到.1 主要结论及其证明定理1 令n ,d 是非负整数,a ,b ,f 是正整数,α,β是两个变量.若方程(3)中的幂级数g λ(t )满足a -d g β(a t )(βf e f a t -1)=b -d g α(b t )(αf e f b t -1)且αa =βb ,则ðni =0n æèçöø÷i a i -d b n -i g (r -1)n -i ,g ,α(a y )ðfa -1j =0g (j )αj g (r )i ,g ,βb x +j b æèçöø÷a =ðni =0n æèçöø÷i b i -d a n -i g (r -1)n -i ,g ,β(b y )ðfb -1j =0g (j )βj g (r )i ,g ,αa x +j a æèçöø÷b (4)①收稿日期:20111008基金项目:国家自然科学基金(11071194);陕西省自然科学基础研究基金(2009J M 1006);西北大学优秀博士学位论文基金(09Y Y B 05).作者简介:何圆(1982),男,四川内江人,博士,讲师,主要从事数论方面的研究.Copyright ©博看网. All Rights Reserved.ðni =0n æèçöø÷i a n -i -d b i g (r )n -i ,g ,β(b x )ðij =0i æèçöø÷j S j ,g ,α(f a )g (r -1)i -j ,g ,α(a y )=ðni =0n æèçöø÷i b n -i -d a i g (r )n -i ,g ,α(a x )ðij =0i æèçöø÷jS j ,g ,β(f b )g (r -1)i -j ,g ,β(b y )(5)证首先证明式(4).将g (m )λm e m t=ðɕn =0g (m )λmm n t nn !(m =0,1, ,k -1)逐项相加,可得g (0)λ0e 0t+g (1)λ1e 1t+ +g (k -1)λk -1e (k -1)t =ðɕn =0S n ,g ,λ(k )t nn !(6)由式(3)和式(6)知ðf a -1j =0g (j )αj ðɕn =0a n -d g(r )n ,g ,βb x +j b æèçöø÷a t n n !=a -d ðɕn =0g (r )n ,g ,β(b x )(a t )n n éëêùûú!ðɕn =0S n ,g ,α(f a )(b t )nn éëêùûú!(7)根据式(7)和C a u c h y 乘积,可得ðɕn =0ðni =0n æèçöø÷i a i -d b n -i g (r -1)n -i ,g ,α(a y )ðfa -1j =0g (j )αj g (r )i ,g ,βb x +j b æèçöø÷éëêêùûúúa t n n !=a -d ðɕn =0g (r -1)n ,g ,α(a y )(b t )nn éëêùûú!ðɕn =0g (r)n ,g ,β(b x )(a t )nn éëêùûú!ðɕn =0S n ,g ,α(f a )(b t )nn éëêùûú!=a -d g β(a t )e a b t (x +y )ðf-1i =0g (i )αi e i b t g α(b t )ðf-1j =0g (j )βj e j a t g β(a t ())r -1ðf a -1i =0g (i )αi e i b tðf-1j =0g (j )βj e j()a t =a -d e a b t (x +y )g β(a t )αf a e f a b t-1αf e f b t -1ðf-1i =0g (i )αi e i b t g α(b t )ðf-1j =0g (j )βj e j a t g β(a t ())r -1ðf-1i =0g (i )αi e i b t ðf-1j =0g (j )βj e j ()a t (8)易知,式(8)中a 和α分别关于b 和β对称,故得式(4).下面证明式(5).根据式(3)㊁式(8)和C a u c h y 乘积,可得ðɕn =0ðni =0n æèçöø÷i a n -i -d b i g (r )n -i ,g ,β(b x )ðij =0i æèçöø÷j S j ,g ,α(f a )g (r -1)i -j ,g ,α(a y éëêêùûúú)t n n !=a -d ðɕn =0g (r )n ,g ,β(b x )(a t )nn éëêêùûúú!ðɕn =0S n ,g ,α(f a )(b t )nn éëêêùûúú!ðɕn =0g (r -1)n ,g ,α(a y )(b t )nn éëêêùûúú!(9)故由式(4)和式(8)知,式(5)是成立的.这便完成了定理1的证明.2 一些应用2.1 高阶A po s t o l -B e r n o u l l i 多项式文献[7]将高阶A p o s t o l -B e r n o u l l i 多项式B (r)n (x ,λ)定义为t λe t-æèçöø÷1r e x t =ðɕn =0B (r )n (x ,λ)t nn !(10)其中r 是非负整数,λ是一个复变量.特别地,r =1的情形给出了A p o s t o l -B e r n o u l l i 多项式B n (x ,λ)和A p o s t o l -B e r n o u l l i 数B n (λ)=B n (0,λ).式(10)中λ=1的情形给出了高阶B e r n o u l l i 多项式B (r)n (x )(参见文献[8]).定理2 令n 是非负整数,a ,b 是正整数,α,β是两个复变量,且αa =βb.则ðni =0n æèçöø÷i a i -1b n -i B (r -1)n -i (a y ,α)ða -1j =0αj B (r )i b x +j b a ,æèçöø÷β=ðni =0n æèçöø÷i b i -1a n -i B (r -1)n -i (b y ,β)ðb -1j =0βj B (r )i a x +j a b ,æèçöø÷α(11)ðni =0n æèçöø÷i a n -i -1b i B (r )n -i (b x ,β)ðij =0i æèçöø÷jS j ,α(a )B (r -1)i -j (a y ,α)=ðni =0n æèçöø÷i b n -i -1a i B (r )n -i (a x ,α)ðij =0i æèçöø÷j S j ,β(b )B (r -1)i -j (b y ,β)(12)其中S n ,λ(k )=λ00k +λ11k + +λk -1(k -1)n .92第4期何圆,等:关于一个多项式序列的对称关系Copyright ©博看网. All Rights Reserved.证在定理1中取f =d =1,g (0)=1和g λ(t )=t λe t -1,定理2即得.注1 定理1中,α=β=1和α=β=λ(λa -1=λb -1=1)的情形分别给出了文献[5]和文献[6]的结果.2.2 高阶A po s t o l -E u l e r 多项式文献[7]将高阶A p o s t o l -E u l e r 多项式E (r)n (x ,λ)定义为2λe t +æèçöø÷1r e x t =ðɕn =0E (r )n (x ,λ)t nn !(13)其中r 是非负整数,λ是一个复变量.特别地,r =1给出了A p o s t o l -E u l e r 多项式E n (x ,λ)和A p o s t o l -E u l e r 数E n (λ)=2n E n (12,λ).式(13)中λ=1的情形给出了高阶E u l e r 多项式E (r)n (x )(参见文献[8]).定理3 令n 是非负整数,a ,b 是正整数,α,β是两个复变量,且αa =βb.则ðni =0n æèçöø÷i a i b n -i E (r -1)n -i (a y ,-α)ða -1j =0αj E (r )i b x +j b a ,-æèçöø÷β=ðni =0n æèçöø÷i b i a n -i E (r -1)n -i (b y ,-β)ðb -1j =0βj E (r )i a x +j a b ,-æèçöø÷α(14)ðni =0n æèçöø÷i a n -i b i E (r )n -i (b x ,-β)ðij =0i æèçöø÷jS j ,α(a )E (r -1)i -j (a y ,-α)=ðni =0n æèçöø÷i b n -i a i E (r )n -i (a x ,-α)ðij =0i æèçöø÷j S j ,β(b )E (r -1)i -j (b y ,-β)(15)证在定理1中取f =1,d =0,g (0)=1和g λ(t )=2-λe t+1,定理3即得.注2 式(15)中α=β=-λ(λa -1=λb -1=1)的情形纠正了文献[6]中结语部分的一个论断.参考文献:[1]L A N GS .C y c l o t o m i cF i e l d s [M ].N e w Y o r k :S p r i n g e r -V e r l a g ,1978.[2] G E S S E L LI .S o l u t i o n t oP r o b l e m E 3237[J ].A m e rM a t h M o n t h l y,1989,96:364.[3] HOWA R DFT.A p p l i c a i o n s o f aR e c u r r e n c e f o r t h eB e r n o u l l iN u m b e r s [J ].JN u m b e rT h e o r y,1995,52:157-172.[4] T U E N T E R HJH.AS y mm e t r y o fP o w e r S u m P o l y n o m i a l s a n dB e r n o u l l iN u m b e r s [J ].A m e rM a t h M o n t h l y ,2001,108:258-261.[5] Y A N GS h e n g -l i a n g .A n I d e n t i t y o f S y m m e t r y f o r t h eB e r n o u l l i P o l y n o m i a l s [J ].D i s c r e t eM a t h ,2008,308(2):550-554.[6] Z HA N GZ h i -z h e n g ,Y A N G H a n -q i n g .S e v e r a l I d e n t i t i e s f o r t h eG e n e r a l i z e dA p o s t o l -B e r n o u l l iP o l y n o m i a l s [J ].C o m p M a t hA p p l ,2008,56:2993-2999.[7] L U O Q i u -m i n g ,S R I V A S T A V A H M.S o m eG e n e r a l i z a t i o n so f t h eA p o s t o l -B e r n o u l l i a n d A p o s t o l -E u l e rP o l y n o m i a l s [J ].JM a t hA n a lA p p l ,2005,308:290-302.[8] N ÖR L U ND NE .V o r l e s u n g e nÜb e rD i f f e r n z e n r e c h n u n g [M ].B e r l i n :S p r i n g e r -V e r l a g ,1924.S y m m e t r i cR e l a t i o n s I n v o l v i n g a S e q u e n c e o fP o l yn o m i a l s H E Y u a n 1,2, Z HA N G W e n -p e n g 21.F a c u l t y o f S c i e n c e ,K u n m i n g U n i v e r s i t y o f S c i e n c ea n dT e c h n o l o g y ,K u n m i n g 650500,C h i n a ;2.D e p a r t m e n t o fM a t h e m a t i c s ,N o r t h w e s t U n i v e r s i t y ,X i a n 710127,C h i n a A b s t r a c t :As e q u e n c e o f p o l y n o m i a l s a s s o c i a t e dw i t h g e n e r a l i z e dB e r n o u l l i p o l y n o m i a l s ㊁h i g h e r o r d e rA p o s -t o l -B e r n o u l l i p o l y n o m i a l s a n dh i g h e r o r d e rA p o s t o l -E u l e r p o l y n o m i a l s a r e s t u d i e d .B y u s i n g c o m b i n a t o r i a l t e c h n i q u e s ,t w o s y mm e t r i c r e l a t i o n s b e t w e e ns u c ha s e q u e n c eo f p o l y n o m i a l s a n ds u m so f p o w e r s a r ee s -t a b l i s h e d .A s a p p l i c a t i o n s ,s o m e c o r r e s p o n d i n g r e s u l t s i n t h e r e f e r e n c e s a r e o b t a i n e d a s s pe c i a l c a s e s .K e y w o r d s :g e n e r a l i z e dB e r n o u l l i p o l y n o m i a l s ;A p o s t o l -B e r n o u l l i p o l y n o m i a l s ;A p o s t o l -E u l e r p o l y n o m i a l s ;03西南师范大学学报(自然科学版) h t t p ://x b b jb .s w u .c n 第38卷Copyright ©博看网. All Rights Reserved.s u m s o f p o w e r s责任编辑廖坤13第4期何圆,等:关于一个多项式序列的对称关系Copyright©博看网. All Rights Reserved.。

Rodrigues Formula for the Nonsymmetric Multivariable Hermite Polynomial

KEYWORDS: nonsymmetric multivariable Hermite polynomial, Calogero model, Dunkl-Cherednik operator formulation, Rodrigues formula
§1.
Introduction
Explicit construction of the commuting conserved operators and identiﬁcation of their simultaneous eigenfunctions are important fundamental problems in the study of quantum integrable systems. The orthogonal basis spanned by the simultaneous eigenfunctions of the conserved operators plays an essential role in detailed studies on the integrable systems such as the calculation of the correlation functions, the representation theory of the symmetry of the system and so on. Among the integrable inverse-square interaction models in one dimension, the Calogero and the Sutherland models1, 2) are considered to be the “twins” because the two models have their own Dunkl-Cherednik operator formulations that share the same structure of the commutator algebra.3, 4, 5, 6) The Dunkl-Cherednik operator formulation provides an explicit construction of the commuting conserved operators of the two models. The celebrated Jack symmetric polynomials7, 8, 9) are the simultaneous eigenfunctions of conserved operators made of the Cherednik operators of the Sutherland model. However, only a little had been known about the symmetric simultaneous eigenfunctions of the conserved operators of the Calogero model that are made of the Cherednik operators.10) Motivated by the Rodrigues formula for the Jack symmetric polynomial that was found by Lapointe and Vinet,11, 12) we presented the Rodrigues formula for the Hi-Jack symmetric (multivariable Hermite) polynomial and identiﬁed it as the simultaneous eigenfunction of the conserved operators of the Calogero model.6, 13, 14) The multivariable Hermite polynomial is a one-parameter deformation of the Jack symmetric polynomial. They share many common properties, which reﬂect the same algebraic structure of the corresponding Dunkl-Cherednik operators. To study the Calogero and Sutherland models including spin variables, we need the nonsymmetric simultaneous eigenfunctions of the Cherednik operators as the orthogonal basis of the orbital part of the wave func∗

PP is closed under intersection

al. 27] have shown that a depth-(d + 1) circuit of threshold gates which com-
putes the parity on n inputs requires (dn1=d=log2n) threshold gates. Recently,
simulated by a circuit with a single threshold gate (at the root) with depth
greater by a constant and only a limited increase in size. If the original circuit
under polynomial-time conjunctive and disjunctive reductions. Consequently,
PP is closed under polynomial-time truth-table reductions in which the truth
Beigel 6], extending our techniques, has shown that that no constant depth cir-
cuit with no(1) threshold gates, 2no(1) AND, OR, and NOT gates, and 2no(1) wires
threshold gates, 2no(1) AND, OR, and NOT gates (in arbitrary positions), and
2no(1) wires can compute parity. This is the rst natural example of a function

Condensation of Glyoxal with Triethylenetetraamine; Isomerization and Cyclization

SHORT COMMUNICATION Condensation of Glyoxal with Triethylenetetraamine;Isomerizationand CyclizationGwe´nae¨lle Herve´,[a]He´le`ne Bernard,[a]Loı¨c Toupet,[b]and Henri Handel*[a] Keywords:Glyoxal/Triethylenetetraamine/Diethyl Oxalate/Cyclen/Ring closure/IsomerizationThe condensation of triethylenetetraamine with glyoxal four compounds exhibit equilibrium and isomerization leads to the formation of four stereoisomers A,B,C,D.A isphenomena which are the key-feature of this cyclen the thermodynamic product and D the kinetic one.These synthesis.The synthesis and study of1,4,7,10-tetraazacyclododecane solution.Its vicinal configuration was unambiguously es-tablished on the basis of its13C NMR spectrum,which ex-(cyclen)has attracted considerable attention owing to its re-markable complexing properties,which have allowed one to hibited only one kind of aminal carbon.The trans relation find medicinal applications of its derivatives.[1]In a previousbetween the two aminal hydrogen atoms was deduced from paper we described a new route to cyclen from the key inter-temperature-dependent13C NMR studies.As no exchangephenomena were observed,we could deduce that this com-mediate bis-aminal obtained by condensation of triethylenete-traamine with aqueous glyoxal.[2]We found that this reaction pound was rigid and totally symmetric;consequently,it led to the four configurations depicted in Scheme1.corresponded to the isomer D.On the other hand,its dia-stereoisomeric vic-structure with a cis-configuration,theisomer C,was conformationally labile and exhibited ex-change phenomena.[2][3]At low temperature,the13C NMRspectrum of C showed a set of eight signals indicating anunsymmetrical configuration.In order to extend the field of applications of these inter-mediates,we explored the cyclization reaction with otherreactants such as diesters,and,more particularly,the di-Scheme1.Bis-aminals(222)-glyoxalethyloxalate.[4]When diethyloxalate was added to a solution We also showed that an irreversible isomerization oc-of the bis-aminal A in ethanol at room temperature,no curred when the mixture of the four isomers was heated in cyclization reaction occurred;when the mixture was heated ethanol in the presence of a small amount of water,leading for several hours only polymeric products were obtained. mainly to the thermodynamic product A which presented However,at room temperature,we observed a reaction with a gem configuration favorable to its cyclization with1,2-the recristallized bis-aminal D leading to compound1in dibromoethane.[2]Isomer D was identified as the kinetic60%yield(Scheme2).product,the major compound obtained when the reactionwas performed at relatively low temperature(Ϫ10°C).Wereported that the vic configuration of D was not favorableto its cyclization with1,2-dibromoethane.This conden-sation leads to polymers.So,prior to the addition of reac-tant,an isomerization to A is necessary.We also noticedthat isomer B was always present in small amounts,lessScheme2.Condensation of D with diethyl oxalatethan10%,and that isomer C never exceeded20%.We wish to report here new investigations on the iso-The structure of this unique reaction product was estab-merization process and another synthesis of cyclen.lished from an X-ray analysis(Figure1).Surprisingly the Starting from the crude mixture of the four isomers,ob-ORTEP-type plots showed that1has two hydrogen atoms tained after glyoxal condensation of triethylenetetraamine in a cis-configuration.So,we wondered whether compound in ethanol atϪ10°C,we observed that only one product1resulted from a rearrangement process occurring during crystallized in75%yield from a tetrahydrofuran/pentane the reaction course,or alternatively,from a previous iso-merization of D.To get an answer to this question we stud-[a]Laboratoire de Chimie,Electrochimie Mole´culaires et Chimie ied the behavior of D in solution.Analytique,UMR6521,Universite´de Bretagne Occidentale,In a first experiment,the isolated isomer D was allowed 6Av.Victor le Gorgeu,F-29285Brest,FranceFax:(internat.)ϩ33-2/98016594to stand in absolute ethanol at room temperature overnight.E-mail:henri.handel@univ-brest.fr No isomerization occurred.The same experiment was car-[b]G.MC.M.Universite´de Rennes I,UMR6626,Baˆt.11A,Campus de Beaulieu,F-35042Rennes,France ried out in95°ethyl alcohol;a DǞC transformation in aSHORT COMMUNICATIONG.Herve´,H.Bernard,L.Toupet,H.HandelFigure1.Molecular structure of compound1in the crystal80:20ratio was reached in fifteen hours,although A wasnot formed.In another experiment,D was refluxed for sev-eral hours in absolute ethanol,and the DǞC transform-ation was observed in the same proportion.After addinga few drops of water and heating the mixture overnight,compound A was obtained as the major isomer(75%).The constancy of the D/C ratio indicated an equilibriumScheme3.Isomerization processbetween these two isomers.These results highlighted the im-portance of water catalysis in the isomerization process.Infact,two steps have to be considered:the first one wasrather fast in the presence of water and occurred at roomtemperature leading to an equilibrium between D and C;the second one,corresponding to the formation of A,re-quired heating and water catalysis.This configurationcorresponds to the most stable product,which contains amaximum number of six-membered rings.In anhydrousplexation of D with Mo(CO)6conditions at room temperature,the formation of A was notobserved and the D v C equilibrium was slowed.Whenstarting from isomer A,no condensation with diethyloxa-bands(2A1ϩB1ϩB2)for a structure possessing a localC2v symmetry were observed.All the carbon atoms were late occurred;so,one can deduce that this second step isnot reversible or must be extremely slow.identified by13C NMR spectroscopy.This was consistent The fact that water was not required for the D v C equi-with the unsymmetrical cis-configuration of bis-aminal rig-librium could be explained by an intramolecular rearrange-idified by its coordination to the metal.The four distinct ment involving an iminium ion as intermediate[5]and facili-signals of equivalent intensity observed between200and tated in protic medium by a proton transfer as depicted in225ppm were attributed to the four carbonyls of the com-plex.[6]Consequently,the D v C equilibrium occurred Scheme3.The formation of compound A is probably the conse-upon warming,even in a strictly anhydrous medium,to give quence of glyoxal release by bis-aminal.We noticed that thea configuration more favorable to the metal coordination. A/B ratio remained constant,which is indicative of a pos-Isomer A did not react under these drastic experimental sible equilibrium between A and B.conditions:no reaction occurred and the reactants were en-So,the condensation of D with diethyloxalate in ethanol tirely recovered.Moreover,this indicated that the(D v C) displaced the D v C equilibrium and the cyclization prod-ǞA transformation was not reversible.uct1was obtained in good yield.An analogous isomeriz-Among the four configurations,only C and A gave rise ation was observed when D was allowed to react with mol-to a cyclic adduct,the first one with diethyloxalate and mol-ybdenum hexacarbonyl in n-dibutyl ether at140°C.Under ybdenum hexacarbonyl and the second one with dibromo these conditions,the complex2was isolated in75%yieldderivatives,respectively,while isomers D and B,both totally (Scheme4).rigid,were unreactive or led to polymers.This behavior canbe explained by the good fitting between the substrate and The IR spectrum of2was consistent with the presenceof a cis-M(CO)4moiety since the four expectedν(CO)the reactive.Indeed,C and A had in common flexible andCondensation of Glyoxal with TriethylenetetraamineSHORT COMMUNICATION([D 8]toluene,75MHz,373K):isomer A :δϭ42.9,51.1,51.3(CH 2-adaptable conformations.One can assume that the template α-N),66.9,77.8(N C N).Isomer B :δϭ45.7,50.5,51.3(CH 2-α-N),effect induced by the formation of the bis-aminal in a cis -71.7,88.9(N C N).Isomer C :δϭ43.8,48.8,51.0(CH 2-α-N),77.2configuration maintained the two secondary nitrogens in a (N-C -N).Isomer D :δϭ44.4,48.8,51.9(CH 2-α-N),80.4(N-C -N).configuration favorable to a cyclization process while pre-Typical Procedure for the Synthesis of 1:The recrystallized vic-trans serving the adaptability required for a further intramolecu-isomer D (700mg.,4.1mmol)was dissolved in ethanol (20mL)lar reaction.and diethyloxalate (557µL,4.1mmol)added.A white precipitate Acid hydrolysis of compound 1led to rearrangement was formed,then filtered after completion of the reaction (15h)products;dioxocyclen and further cyclens were not ob-and dried in vacuo to give pure 1in 60%yield (450mg),m.p ϭtained this way.However,the reduction of the two amide 252°C.Ϫ1H NMR (CDCl 3,400MHz,298K):δϭ4.12(N-C H -functions gave easily the previously described cyclen-glyoxal C H -N):4.12ppm (s).Ϫ13C NMR (CDCl 3,100MHz,298K):δϭadduct,[8]which was subsequently treated with hydrazine 44.4,47.0,49.6(C H 2-α-N);69.7(N-C -N),157.5(C O).Ϫmonohydrate to give the cyclen deprotection as previously C 10H 14N 4O 2:calcd.C 54.05,H 6.30,N 25.22;found:C 53.81,H described (Scheme 5).[2]6.52,N 25.43.ϪHRMS m /z ϭ223.1195[M ϩH]ϩ.Crystallographic data (excluding structure factors)were depositedwith the Cambridge Crystal Data Centre as supplementary publi-cation DC-118129.Copies of the data can be obtained free of charge on application to CCDC,12Union Road,Cambridge CB2,1EZ,UK.[Fax:(internat.)ϩ44Ϫ1223/336Ϫ033;E-mail:deposit@].Typical Procedure for the Synthesis of 2:Complexation was carried Scheme 5.Reduction of 1and deprotectionout under a nitrogen atmosphere using standard Schlenk tech-niques.Solvents were freshly distilled from an appropriate desicc-In summary,the synthesis of the bis-aminal triethylenete-ant (P 2O 5for Bu 2O,CaH 2for hexane).Sublimed Mo(CO)6traamine-glyoxal involved both thermodynamic and kinetic (290mg,1.1mmol)and bis-aminal D (168mg,1.00mmol)were phenomena leading to the formation of four isomers.The heated under reflux (142°C)in n -dibutyl ether (20mL)for 3h,isomer D ,the kinetic compound,was obtained as the major while occasionally returning the sublimed Mo(CO)6to the reaction component of the mixture when the reaction was con-solution by scraping the condenser walls.A yellow precipitate was trolled.Concerning thermodynamics,we have seen the im-formed during the reaction.After cooling to room temperature,the solid was separated,washed with hexane (3ϫ20mL)and then portant part played by water and/or heating in the product dried in vacuo at 50°C to yield 2in 75%.ϪIR (CH 2Cl 2):2000transformations.Cyclen was obtained through a controlled (w),1880(s,sh),1865(vs),1827(m)ν(CO)cm Ϫ1.Ϫ13C NMR isomerization process.([D 6]DMSO,75MHz,298K):δϭ47.5,48.0,48.3,51.7,54.2,54.6The advantage of this method lies in the easy purification (C H 2-α-N),73.8,82.0(N-C -N),206.5,207.6,220.7,222.0(C O).of the intermediate.No by-products are obtained in the two Reduction of 1,Hydrolysis and Deprotection:To a mixture of 1in next steps (reduction and deprotection).Thus,this reaction THF (30mL)was added an excess of BH 3·SMe 2(5/3equivalents)sequence constitutes a clean and facile new route to cyclen.and the mixture refluxed for 48h.After cooling,the unreacted BH 3·SMe 2was destroyed by slow addition of methanol,and the solution evaporated to leave a white solid.This was taken up in 10%Experimental Sectionaqueous HCl (30mL)and refluxed overnight.After cooling,the pH All the NMR spectra were recorded with a Bruker AC 300(13C:was raised to 14with NaOH pellets and the product extracted with 100.62MHz,1H:300MHz)or with a Bruker Advance DRX 400CH 2Cl 2(3ϫ30mL).After drying (MgSO 4)and solvent evapor-(13C:75.47MHz,1H:400MHz)spectrometer.Chemical shifts δation,the well-known cyclen-glyoxal was isolated in 60%yield.The (in ppm)are given relative to the solvent.IR spectra were recorded deprotection step of cyclen-glyoxal was achieved by refluxing in hy-with a Perkin-Elmer 1430spectrometer.High Resolution Mass drazine monohydrate as described elsewhere.[2]Ϫ13C NMR (CDCl 3,spectra were recorded with ZabSpecETOF EI ϩVG analytical 100MHz,298K):δϭ50.0,50.7(C H 2-α-N),78.8(N-C -N).spectrometer.Elemental analyses were carried out by the Centre de Microanalyses du CNRS de Lyon,France.[1][1a]Schering A -G.Ger.Offen .,DE 19,652,386,1998.Ϫ[1b]Scher-ing A -G.Ger .Offen .,DE 19,652,387,1998.Ϫ[1c]International Typical Procedure for the Synthesis of Bis-aminals [(222)-glyoxal]:Patent,Bracco S.P .A,N °WO 99/05128,1999.Ϫ[1d]E.Kimura,Triethylenetetraamine (2.00g,13.69mmol)was dissolved in etha-T.Ikeda,M.Shionoya,M.Shiro,Angew.Chem.Int.Ed.Engl .nol (30mL ϩ2drops of glacial acetic acid).The resulting solution 1995,34,663.was cooled to Ϫ10°C and a cooled solution of glyoxal (40%in [2]G.Herve´,H.Bernard,N.Le Bris,M.Le Baccon,J.-J.Yaouanc,H.Handel,Tetrahedron Lett .1999,40,2517.water,1.98g,13.69mmol)in ethanol was added dropwise.After [3]J.Jazwinski,R.A Kolinski,Tetrahedron Lett .1981,22,pletion of the reaction (2h),an excess of K 2CO 3was added.[4]J.W .Krajovski,P .Gluzinski,R.A Kolinski,A.Kemme,A.Filtration and solvent evaporation under reduced pressure gave a Mishnev,Polish J.Chem .1994,68,703.clear oil,which was taken up in toluene (20mL).The mixture was [5]K.Janne´,P .Ahlberg ,J.Chem.Res.(S)1977,286.[6]G.Herve´,N.Le Bris,H.Bernard,J.-J.Yaouanc,H.des Ab-allowed to stand 15min and then polymers were eliminated by fil-bayes,H.Handel ,anomet.Chem .1999,2,259.tration.The filtrate was evaporated and the procedure repeated [7]G.R.Weisman,S.C.H.Ho,V .Johnson,Tetrahedron Lett .twice to give pure bis-aminal in 90%yield (2.08g)as a mixture 1980,21,335.of the four isomers.Recrystallization of D from tetrahydrofuran/Received May 21,1999[O99306]pentane yielded white crystals in 75%(1.55g).Ϫ13C NMR。

齐次多项式英语

齐次多项式英语Homogeneous Polynomials in EnglishPolynomials are a fundamental concept in mathematics that play a crucial role in various areas of study, including algebra, calculus, and beyond. Among the different types of polynomials, homogeneous polynomials hold a unique and significant place. In this essay, we will delve into the world of homogeneous polynomials, exploring their definition, properties, and applications.A polynomial is a mathematical expression that consists of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. Homogeneous polynomials are a special class of polynomials where all the terms have the same degree, or the sum of the exponents of the variables in each term is the same. This means that in a homogeneous polynomial, the degree of each term is equal to the degree of the entire polynomial.For example, consider the polynomial expression: 3x^2 + 2xy + y^2. This is a homogeneous polynomial of degree 2 because the degree of each term (the sum of the exponents of the variables) is 2. In contrast, the polynomial 2x^2 + 3xy + 4y is not homogeneous, asthe degree of the first term is 2, while the degree of the second and third terms is 1.One of the fundamental properties of homogeneous polynomials is their behavior under scalar multiplication. If a homogeneous polynomial is multiplied by a scalar (a constant), the resulting polynomial is also homogeneous, and its degree remains the same as the original polynomial. This property is known as the homogeneity property and is mathematically expressed as follows: if f(x) is a homogeneous polynomial of degree n, then f(tx) = t^n f(x), where t is a scalar.This homogeneity property has numerous applications in various fields of mathematics and science. In physics, for example, it is used in the study of dimensional analysis, where homogeneous polynomials are employed to ensure the consistency of units and dimensions in physical equations. In economics, homogeneous functions, which are closely related to homogeneous polynomials, are used to model production functions and study the behavior of firms and industries.Moreover, homogeneous polynomials play a crucial role in the field of algebraic geometry, where they are used to define and study algebraic varieties, which are the geometric objects represented by polynomial equations. Homogeneous coordinates, a fundamentalconcept in projective geometry, rely heavily on the properties of homogeneous polynomials.In the realm of pure mathematics, homogeneous polynomials are studied for their intrinsic mathematical properties and their connections to other areas of mathematics. For instance, the theory of invariants, which deals with the properties of polynomial expressions that remain unchanged under certain transformations, is closely linked to the study of homogeneous polynomials.Furthermore, homogeneous polynomials are essential in the study of differential equations, where they are used to construct solutions and analyze the behavior of dynamical systems. In the field of optimization, homogeneous polynomials are employed in the formulation and analysis of optimization problems, particularly in the context of convex optimization.In conclusion, homogeneous polynomials are a fascinating and important class of mathematical objects that have far-reaching applications in various disciplines. Their unique properties, such as the homogeneity property, have made them indispensable tools in the arsenal of mathematicians, scientists, and engineers. As we continue to explore the depths of mathematics and its applications, the study of homogeneous polynomials will undoubtedly remain acrucial area of inquiry, contributing to our understanding of the world around us.。

Confined Polymers Big World in a Small Space密闭的聚合物大世界在一个小空间共33页

Confined Polymers: unity of thought
* in the test tube
* in polymer brushes
* in mesoscopic channels
Adrian Parsegian and many friends: Sergey Bezrukov, Joel Cohen, Per Lyngs Hansen,
M a c in to s h P IC T im a g e fo rm a t
is n o t s u p p o rte d
~ 1, crudely speaking
cp = molar concentration Cp = weight concentration Mm = monomer weight
5 10 15 20 25 30
PEG 3400, weight %
MW 3400 ~ 77 monomers of length 3.4 Angstroms Stretches to ~ 28 nm (vs. ~10 nm length of pore) Rg ~ 2.5 nm in dilute solution ( vs. ~ 1 nm radius of pore)
Test tube
~cm
vs.
~nm
grafted,
between bilayers
vs. inside a pore
~10 nm ~nm
Schematic of force measurement between PEG lipids
~ cm
Posmotic
Equilibrate in high-concentration 9/4-power limit

结构化学分子对称性

reaction coordiants
群的表示
Symmetry operations obey the laws of group theory.
A symmetry operation can be represented by a matrix operating on a base set describing the molecule. Different basis sets can be chosen, they are connected by similarity transformations. S-1AS diagonal block matrix For different basis sets the matrices describing the symmetry operations look different. However, their character (trace) is the same!
NO
Special Group?
YES
28
分子的偶极矩和极化率
偶极矩：dipole, μ=qr (库仑米 Cm)
μ=4.8×10-18 cm esu =4.8D Debye 1D=3.336×10-30 Cm
只有属于Cn和Cnv 这两类点的分子才可能具有永久偶极矩诱导偶极矩：在电场E中分子发生诱导极化而产生的

a)具有对称中心的
b)没有对称中心的 1 0 0 i 0 1 0 0 0 1
E, n i i,
n is even n is odd
a)氨分子（NH3）的三重轴
b)水分子（H2O）的二重轴
x x' k Cn y y ' z z'

关于COOK算例的数值解

关于COOK算例的数值解胡圣荣;喻菲菲【摘要】COOK算例没有理论解，文献中常用的参考解与大量优秀单元的计算解有较明显差别，影响了对单元性能的客观公正评价。

为获得更准确、更可靠的计算解，分别用4-40次高次元和Q8、T6、QM6低次元进行逼近，考察算例的有限元渐进收敛解。

结果表明，在5位有效数字下该问题的解为vC=23.967，σAmax=0.23687，σBmax=-0.20352。

%With no theoretical solution found with Cook’s problem,numerical solution is used for it,but the commonly used one often shows obvious difference with those of large number of outstanding elements,which affects objective and fair evaluation to the element performance. In order to obtain more accurate and reliable numerical solution,high-order elements of 4-40 order and low-order elements of Q8,T6,Qm6 are used to investigate this problem’s progressive convergence solution respectively. Results show that the problem’s solution in five significant digits i svC=23.967,sAmax=0.23687,sBmin=-0.203 52.【期刊名称】《广州城市职业学院学报》【年(卷),期】2015(000)002【总页数】4页(P54-57)【关键词】COOK问题;COOK梯形梁;单元性能测试【作者】胡圣荣;喻菲菲【作者单位】华南农业大学工程基础教学与训练中心，广东广州510642;广东技术师范学院机电学院，广东广州510635【正文语种】中文0 引言单元性能对有限元分析结果至关重要，在单元性能的测试中，Cook 梯形梁是一个常用的算例，见图1。

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International Electronic Journal of AlgebraVolume2(2007)208-225D-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS OVER INTEGRAL DOMAINS D OF ANY CHARACTERISTICJonathan GrovesReceived:6April2007;Revised:11June2007Communicated by Derya Keskin T¨u t¨u nc¨uAbstract.Let D be any integral domain of any characteristic.A polynomialp(x)∈D[x]is D-nice if p(x)and its derivative p (x)split in D[x].We give acomplete description of all D-nice symmetric polynomials with four roots overintegral domains D of any characteristic not equal to2by giving an explicitformula for constructing these polynomials and by counting equivalence classesof such D-nice polynomials.To illustrate our results,we give several exampleswe have found using our formula.We conclude by stating the open problemofﬁnding all D-nice symmetric polynomials with four roots over integral do-mains D of characteristic2and all D-nice polynomials with four roots over allintegral domains D of any characteristic.Mathematics Subject Classiﬁcation(2000):Primary:13F20;Secondary:11C08,26C10Keywords:Critical point,D-nice polynomial,integral domain,nice polyno-mial,polynomial,root,symmetric polynomial1.IntroductionLet D be any integral domain of any characteristic.We say that a polynomial p(x)∈D[x]splits in D[x]if,for some n≥0,p(x)can be written in the form p(x)=a(x−r1)(x−r2)···(x−r n)where a=0and a,r1,...,r n∈D.We say that p(x)∈D[x]is D-nice or(nice in D)if p(x)and its derivative p (x)split in D[x].By our deﬁnition of splitting,if p(x)is D-nice,then p (x)=0.Most mathematicians who had researched Z-nice polynomials were interested in constructing polynomials with integer coeﬃcients,roots,and critical points—polynomials that are“nice”for calculus students to sketch(see[1],for example).We found this problem of extending these earlier results in Z to all integral domains D of any characteristic worth further study.Since many earlier papers use the term nice instead of Z-nice,we too will often use the term nice instead of Z-nice.Theﬁrst paper to consider nice polynomials[4]D-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS209was published in1960.This paper,along with the papers[1]and[12],give a formula for constructing all nice cubics.The paper[3]gives theﬁrstﬁve known examples of nice nonsymmetric quartics and a formula for all nice symmetric quartics.In1989 the problem ofﬁnding all nice quartics using elliptic curves was added to the list of unsolved problems[10]in the American Mathematical Monthly.Ten years later, the problem of constructing,describing,and classifying nice polynomials was added to the the list of unsolved problems[11]in the American Mathematical Monthly. Other papers soon followed,including the main paper on nice polynomials[2],the submitted paper[5]with a diﬀerent approach to D-nice polynomials,the accepted paper[6]with several new results on nice symmetric and antisymmetric polyno-mials,the accepted paper[7]with a complete description of all nice polynomials with three distinct roots(which generalizes the known results on nice cubics),and the accepted paper[8]with a complete description of all nice symmetric polynomi-als with four distinct roots(which generalizes the known results on nice symmetric quartics).The paper[5]takes a new approach to D-nice polynomials by considering the relations between the roots and critical points of all polynomials over integral domains D of characteristic0.The main result in[5]gives several properties of D-nice polynomials p(x)where D is an integral domain of characteristic0satisfying D∩Q=Z and where the degree of p(x)is a prime power in D[5,Theorem6.3]. All the results in[5],including the main one,and all the results in[6]-[8]follow from these relations.The relations as stated in[5]and[6]depend on the degree of p(x),not on the maximal number of distinct roots of p(x).The relations in[7] and[8]given for polynomials with up to three and four distinct roots,respectively, do not depend on the degree of the polynomials.The latest paper[9]gives the relations between the roots and critical points of all polynomials p(x)∈D[x]where D has any characteristic and where the relations depend on the maximal number of distinct roots of p(x),not on the degree of p(x).The only results on nice polynomials of degree higher than5that has been published so far are[2,Theorem2]and the formula for nice symmetric quartics in [3].But most of the results in[5]-[9]apply to nice polynomials of degrees higher than5.Furthermore,[5]and[9]are the only papers besides this one that consider integral domains other than Z or Q.We now solve the problem of constructing all D-nice symmetric polynomials with four distinct roots(where D has any characteristic other than2).This is a natural generalization of the problem in[8]ofﬁnding all nice symmetric polynomials with four distinct roots.We explain later why this problem remains unsolved for integral210JONATHAN GROVESdomains of characteristic2.Weﬁnd an explicit formula for constructing D-nice symmetric polynomials with four roots,and we count equivalence classes of such D-nice polynomials.We begin by deﬁning the type of a polynomial and what it means for two D-nice polynomials to be equivalent.2.PreliminariesThe type of a polynomial is a list of the multiplicities of its distinct roots.For example,all polynomials of the type(6,5,5,3)are of the form p(x)=a(x−r1)6(x−r2)5(x−r3)5(x−r4)3where r1,r2,r3,and r4are all distinct and a=0.Most of the earlier papers on nice polynomials note that horizontal translations by integers,horizontal or vertical stretches by integer factors,and reﬂections over the coordinate axes transform a nice polynomial p1(x)into another nice polynomial p2(x).Each of these transformations has an inverse transformation that transforms p2(x)into p1(x).The paper[5]extends these transformations and their inverses to all integral domains D of characteristic0[5,Proposition2.1and Corollary2.2]. But it is easy to extend these to integral domains of characteristic p>0.For convenience,we will refer to these transformations by using the same geometric descriptions we use in Z or in Q,even if D is not ordered.Thus,we deﬁne the horizontal translation of p(x)by a∈D units to be p(x−a).The horizontal stretch and compression of p(x)by a nonzero factor of a∈D are deﬁned by p(x/a)and p(ax),respectively.If necessary,the division occurs in theﬁeld of fractions of D. The vertical stretch and compression of p(x)by a nonzero factor of a∈D arep(x),respectively.Reﬂections of p(x)over the x-and y-axes deﬁned by ap(x)and1aare deﬁned by−p(x)and p(−x),respectively.All these transformations preserve the type of a polynomial.Two newly discovered transformations that behave similarly are the power trans-formation and its inverse,the root transformation[6,Theorem2.1]:For any natural number n,a polynomial p(x)is nice iﬀ[p(x)]n is nice.This result clearly holds in any integral domain D of characteristic0and holds in integral domains of characteristic[p(x)]n=0; p>0as long as n is not a multiple of p.If n is a multiple of p,then ddxso,by our convention,[p(x)]n is not D-nice.It is obvious that the root transforma-tion transforms a D-nice polynomial p1(x)into another D-nice polynomial p2(x) iﬀp1(x)=[p2(x)]n for some natural number n and some D-nice polynomial p2(x). The power transformation and the root transformation do not preserve the type ofD-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS211a polynomial.More precisely,if p(x)is of the type(m1,...,m s),then[p(x)]n is of the type(nm1,...,nm s).Translating horizontally a∈D units,stretching horizontally or vertically by factors of a∈D,reﬂecting over the coordinate axes,and taking powers are trans-formations we call equivalence tranformations.The corresponding inverse trans-formations are translating horizontally−a∈D units,compressing horizontally or vertically by factors of a∈D,reﬂecting over the coordinate axes,and taking roots.Although equivalence transformations do transform a D-nice polynomial into a D-nice polynomial,these inverse transformations do not necessarily transform a D-nice polynomial into another D-nice polynomial.For example,a horizontal or vertical compression may transform a D-nice polynomial p1(x)into a polynomial p2(x)where p2(x)is nice in theﬁeld of fractions of D rather than in D.The root transformation applied to arbitrary D-nice polynomials may result in nonpolyno-mials.Since anyﬁnite composition of equivalence tranformations transform a D-nice polynomial p1(x)into another D-nice polynomial p2(x),we say that the two D-nice polynomials p1(x)and p2(x)are equivalent whenever p1(x)can be transformed into p2(x)and vice-versa by aﬁnite composition of equivalence transformations or their inverse transformations.Since all equivalence transformations and their inverses, except the power transformation and its inverse,preserve the type of a polynomial, when we count equivalence classes of D-nice polynomials,we count the number of equivalence classes of D-nice polynomials of the same type.We note that,by the power transformation and its inverse the root transformation,the number of equivalence classes of D-nice polynomials of the type(m1,...,m s)is the same as the number of equivalence classes of D-nice polynomials of the type(nm1,...,nm s) for any n>1if D has characteristic0and any n>1that is not a multiple of p if D has characteristic p>0.In fact,we may consider these types to be the“same”type.A polynomial p(x)∈D[x]is symmetric if there exists a unique c in some exten-sionﬁeld of D,called the center,such that p(c−x)=p(c+x)for all x,and p(x) antisymmetric if p(c−x)=−p(c+x)for all x.Weﬁrst prove that the center c of a D-nice symmetric or antisymmetric polynomial p(x)lies in D.We then use this result to make an assumption that greatly simpliﬁes our work.212JONATHAN GROVESProposition2.1.Let D be any integral domain of any characteristic,and let p(x) be a D-nice symmetric or antisymmetric polynomial with(unique)center c.Then c∈D.Proof.Suppose p(x)is a D-nice antisymmetric polynomial with center c,so p(c−x)=−p(c+x)for all x.Let x=0,so p(c)=−p(c).If D has characteristic not equal to2,then p(c)=0,so c∈D.Now suppose p(x)is a D-nice symmetric polynomial with center c.Then p(c+ x)=p(c−x)for all x.Diﬀerentiating both sides of this equation gives−p (c+x)= p (c−x),so p (x)is antisymmetric with center c.As shown above,if D has characteristic not equal to2,then p (c)=0,so c∈D.Finally,if D has characteristic2,then p(x)=p(−x)=−p(x)for all p(x),so any polynomial(whether D-nice or not)is symmetric or antisymmetric with center0. Since0∈D,the proof is complete.By Proposition2.1,we may center any D-nice symmetric polynomial at0.Since the translated polynomial is equivalent to the original one,we may assume that all D-nice symmetric polynomials have center0.Multiplying p(x)by a nonzero element of D(which is a vertical stretch or compression)results in an equivalent D-nice polynomial,so we may assume p(x)is monic.These two assumptions greatly simplify the problem of constructing and counting equivalence classes of D-nice symmetric polynomials with four distinct roots.Thus,with these assumptions, such polynomials have the form p(x)=(x2−r21)m1(x2−r22)m2if the characteristic of D is not equal to2.3.The Case of Integral Domains D of Characteristic0Weﬁrst consider the problem of constructing and counting equivalence classes of all D-nice symmetric polynomials with four distinct roots over integral domains D of characteristic0.To construct these,we use Lemma3.1below,which gives the relations between the roots and critical points of all symmetric polynomials in QF(D)[x]with four roots where QF(D)is theﬁeld of fractions of D.As we will see below,this set of relations is the key tool inﬁnding our formula for constructing these D-nice symmetric polynomials.Since it is easier toﬁnd a formula that gives QF(D)-nice symmetric polynomials than one that gives D-nice symmetric polyno-mials,we state Lemma3.1in terms of QF(D)-nice symmetric polynomials.When weﬁnd a QF(D)-nice symmetric polynomial with our formula,we may stretch itD-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS213horizontally to obtain an equivalent D-nice symmetric polynomial.Thus,our for-mula may be used to construct and count equivalence classes of D-nice symmetric polynomials of the same type with four distinct roots.The paper[8]takes the same approach in constructing and counting equivalence classes of nice symmetric polynomials with four roots using the formula for Q-nice symmetric polynomials with four roots.Lemma3.1below follows directly from[9,Corollary4.3],which gives the relations between the roots and critical points of all symmetric poly-nomials p(x)∈D[x]with an even number of roots.Lemma3.1in[8]gives the relations between the roots and critical points of all polynomials in Q[x]with four ter,in the same paper,these relations are used to derive the relations for symmetric polynomials in Q[x]with four roots[8,Lemma4.1].We mention this because Lemma3.1below may be proven without using[9,Corollary4.3]by using arguments similar to the ones used in[8].Lemma3.1.[Key lemma.]Let D be an integral domain of characteristic0,and let QF(D)be theﬁeld of fractions of D.Let p(x)=(x2−r21)m1(x2−r22)m2∈QF(D)[x]be a symmetric polynomial of degree d=2m1+2m2with derivative p (x)=dx(x2−r21)m1−1(x2−r22)m2−1(x2−c2)and with four roots in QF(D). Then p(x)is QF(D)-nice iﬀthere exists a number c in QF(D)such thatm2r21+m1r22=(m1+m2)c2.(3.1)Remark.By Lemma3.1,all solutions to(3.1)in QF(D)give us all QF(D)-nice symmetric polynomials of the type(m1,m1,m2,m2).Thus,toﬁnd our formula, we solve(3.1)in QF(D).Our formula is stated in the following theorem.Theorem3.2.[Main theorem:The formula for constructing D-nice symmetric polynomials with four roots.]Let D be an integral domain of characteristic0,and let QF(D)be itsﬁeld of fractions.The polynomial p(x)=(x2−r21)m1(x2−r22)m2∈QF(D)[x]of degree d=2m1+2m2with derivative p (x)=dx(x2−r21)m1−1(x2−r22)m2−1(x2−c2)is QF(D)-nice iﬀc=m2a2+m1b22m2a+2m1b,(3.2)r1=c−a,(3.3)r2=c−b(3.4)where a and b are nonzero elements of QF(D)such that2m2a+2m1b=0or, equivalently,a=(−m1m2)b.214JONATHAN GROVESProof.Set r1=c−a and r2=c−b where a,b∈D are nonzero.Since D has characteristic0,r1=c and r2=c,so a and b are nonzero.Substituting these expressions for r1and r2into(3.1)and simplifying the equation gives the equation m2a2+m1b2=(2m2a+2m1b)c.Solving this for c gives the formula above.To complete the proof,we prove that if a and b are nonzero and if a=(−m1/m2)b, then the relation(3.1)has no solution.Again let r1=c−a and r2=c−b to ob-tain the equation m2a2+m1b2=(2m2a+2m1b)c.Now assume a=(−m1/m2)b. Substituting into our equation,we have m2[(−m1/m2)b]2+m1b2=0,or(m21/m2+ m1)b2=0,which is a contradiction since b=0and m21/m2+m1=0.The converse follows from Lemma3.1.Remarks.(1).The formulas in[8]for nice symmetric polynomials with four roots are similar to formulas(3.2)-(3.4)but are not exactly the same as these.The proof in[8]assumes r1<r2and then uses Rolle’s theorem for R to conclude r1<c<r2. Thus,Rolle’s theorem guarantees the existence of a,b>0such that r1=c−a and r2=c+b;but,for arbitrary integral domains D,we cannot use Rolle’s theorem to establish the existence of such an a and b since D is not necessarily ordered. However,we may conclude such a nonzero a and b exist as we did in the proof above,but all we may say now is that r1,r2,and c are all distinct.Thus,formulas (3.2)-(3.4)are not exactly the same as the formulas in[8].(2).Since Theorem3.2is an equivalence,we mayﬁnd representatives of all the equivalence classes of D-nice symmetric polynomials of the type(m1,m1,m2,m2) by choosing nonzero a and b so that a=(−m1m2)b and using formulas(3.2)-(3.4) above.In this sense,our formulas give all examples of D-nice symmetric polyno-mials of the type(m1,m1,m2,m2).We now give two examples of Z[i]-nice symmetric polynomials we have found using formulas(3.2)-(3.4).Example3.3.Suppose m1=2and m2=5.Let a=2i and b=1+i.Thenthe formulas give c=137+3137i,r1=137−4337i,r2=−3637−637i.Stretching ourpolynomial horizontally by a factor of37,we see that our polynomial is equivalent to the following Z[i]-nice polynomial:p(x)=[x2−(1−43i)2]2[x2−(36+6i)2]5,(3.5)p (x)=14x[x2−(1−43i)2][x2−(36+6i)2]4[x2−(1+31i)2].D-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS215 Example3.4.Suppose m1=6and m2=5.Let a=3+i and b=6−i.Then theformulas give c=31981301−4731301i,r1=−7051301−17741301i,r2=−46081301+8281301i.Stretchingour polynomial horizontally by a factor of1301,we see that our polynomial is equivalent to the following Z[i]-nice polynomial:p(x)=[x2−(705+1774i)2]6[x2−(4608−828i)2]5,(3.6)p (x)=22x[x2−(705+1774i)2]5[x2−(4608−828i)2]4·[x2−(3198−473i)2].These two examples are not equivalent to nice polynomials(i.e.,to Z-nice poly-nomials).To see this,ﬁrst note that any two D-nice symmetric polynomials with four roots centered at0are equivalent using only horizontal stretches or compres-sions.Because horizontal stretches and compressions preserve the ratio a/b where a=c−r1and b=c−r2and because nice symmetric polynomials with four distinct roots centered at the origin satisfy a/b∈Q,we see that these two examples are not equivalent to nice polynomials because the ratios a/b/∈Q.We now determine conditions on a and b so that our formulas(3.2)-(3.4)give a D-nice symmetric polynomial with exactly four roots.To do so,weﬁnd all ratios a/b so that the formulas(3.2)-(3.4)fail to give a D-nice symmetric polynomial with exactly four roots.We consider the three cases r1=±r2,r1=0,and r2=0.By the proof of Theorem3.2,c is undeﬁned iﬀa/b=−m1/m2,and no D-nice symmetric polynomials exist when a/b=−m1/m2;so the case where c is undeﬁned has already been considered.Case1.Suppose a and b are chosen so that r1=±r2.We solve the two equations r1=±r2for a in terms of b,which is equivalent to solving the equations c−a=±(c−b).Solving c−a=c−b for a gives us a=b.We now solve c−a=−(c−b) for a,which is equivalent to the equation2c=a+ing(3.2),we now have m2a2+m1b2m2a+m1b=a+b.By clearing fractions,we obtain the equation m2ab+m1ab=0. If a and b are both nonzero(so ab is nonzero),then this last equation is equivalent to m1=−m2,which is a contradiction since D has characteristic0.Thus,a=0 or b=0but not both.Furthermore,it is easy to see that if a=b=0,then r1=r2 and if a=0or b=0(but not both),then r1=−r2.Thus,r1=±r2iﬀa=0 or b=0(but not both)or a=b=0.Thus,we require that a and b be distinct nonzero elements of QF(D).Case2.Suppose a and b are chosen so that r1=0.By(3.3),r1=0iﬀc=a,so we solve the equation c=a for the ratio a/b.Replacing c by the expression in(3.2)216JONATHAN GROVESand clearing fractions gives us the equation m1b2−2m1ab−m2a2=0.We nowdivide both sides by b2=0and rewrite the equation as m2(ab )2+2m1ab−m1=0,where a/b is regarded as a variable.By the quadratic formula,a b =−m1±112m2.(3.7)These steps are reversible,so choosing a/b equal to one of these numbers gives r1=0.Thus,r1=0iﬀa/b is one of the numbers in(3.7).However,r1will never be0if m1(m1+m2)=dm12is not a square in D.Case3.Suppose a and b are chosen so that r2=0.By(3.4),r2=0iﬀc=b,so we solve the equation c=b for the ratio a/b.Proceeding as in Case2,we obtainthe equation m2(ab )2−2m2ab−m1=0.By the quadratic formula,ab=m2±m2(m1+m2)m2.(3.8)These steps are reversible,so choosing a/b equal to one of these numbers givesr2=0.Thus,r2=0iﬀa/b is one of the numbers in(3.8).However,r2will never be0if m2(m1+m2)=dm22is not a square in D.Cases1-3lead to the following proposition.Proposition3.5.[Conditions on a/b so that the formulas give a D-nice symmetric polynomial with exactly four roots.]Assume the hypotheses of Theorem3.2,and suppose nonzero a and b are chosen.Then the formulas(3.2)-(3.4)give exactlyfour roots iﬀthe ratio a/b is not an element of the set{1,−m1m2,R1,R2,R3,R4} where R1and R2are the numbers deﬁned in(3.7)and R3and R4are the numbers deﬁned in(3.8).We now count equivalence classes of D-nice symmetric polynomials of the type (m1,m1,m2,m2)over integral domains D of characteristic0.Weﬁrst note that any two D-nice symmetric polynomials obtained from formulas(3.2)-(3.4)are equivalent iﬀtheir ratios a/b are equal.We state this result more precisely and prove it below. Proposition 3.6.[Necessary and suﬃcient conditions for two D-nice symmet-ric polynomials of the type(m1,m1,m2,m2)to be equivalent.]Let D be an inte-gral domain of characteristic0.Suppose p1(x)=(x2−r21)m1(x2−r22)m2where p 1(x)=dx(x2−r21)m1−1(x2−r22)m2−1(x2−c21)and p2(x)=(x2−s21)m1(x2−s22)m2 where p 2(x)=dx(x2−s21)m1−1(x2−s22)m2−1(x2−c22)are two D-nice symmetric polynomials with four distinct roots.Let a1=c1−r1,b1=c1−r2,a2=c2−s1,and b2=c2−s2.Then p1(x)and p2(x)are equivalent iﬀa1b1=a2b2.D-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS217Proof.Since p1(x)and p2(x)are symmetric,monic,and centered at0,p1(x)and p2(x)are equivalent using only horizontal stretches and compressions.Since hori-zontal stretches and compressions preserve the ratio a/b deﬁned by formulas(3.3)-(3.4)above,the result follows.Corollary3.7.[Number of equivalence classes of D-nice symmetric polynomials of the type(m1,m1,m2,m2)where D has characteristic0.]Let D be an integral domain of characteristic0,and let m1and m2be any two positive integers.Then there exist inﬁnitely many equivalence classes of D-nice symmetric polynomials of the type(m1,m1,m2,m2).,R1,R2,R3,R4}as in Propo-Proof.Fix m1and m2.Deﬁne the set A={1,−m1m2sition3.5.Choose all nonzero ratios a/b∈QF(D)−A,and use formulas(3.2)-(3.4) toﬁnd r1,r2,and c.Then each ratio a/b gives a QF(D)-nice symmetric polynomial with four distinct roots,which is equivalent to a D-nice symmetric polynomial with four distinct ing Proposition3.6and that QF(D)−A is inﬁnite,we see that any two distinct examples we have constructed are not equivalent,so the result follows.4.The Case of Integral Domains D of Characteristic p>2We now consider the analogous problem over integral domains D of characteristic p>2.First note that if D has characteristic2,then p(x)=p(−x)=−p(x),so any p(x)∈D[x]is both symmetric and antisymmetric with center0.Thus,solving the problem of constructing all D-nice symmetric polynomials with four roots where D has characteristic2is the same as solving the problem of constructing all D-nice polynomials with four roots where D has characteristic2.However,the problem of constructing all D-nice polynomials with four distinct roots over any integral domain D,including those of characteristic2,has not been solved,so we assume D has characteristic p>2.We give more details in Section5.The proof of Lemma3.1does not use the assumption that D has characteristic 0,only the assumption that the degree of p (x)is d−1if p(x)has degree d.Thus, Lemma3.1does apply to p(x)=(x2−r21)m1(x2−r22)m2if the degree d of p(x) is not a multiple of p since p (x)has degree d−1,so formulas(3.2)-(3.4)may be used to construct these types of D-nice symmetric polynomials.But we will show in Theorem4.1below that if the degree of p(x)is not a multiple of p but one of the multiplicities m1and m2is a multiple of p,then we need not use formulas (3.2)-(3.4)to construct these types of D-nice symmetric polynomials.However,218JONATHAN GROVESLemma 3.1does not apply to p (x )=(x 2−r 21)m 1(x 2−r 22)m 2if the degree d of p (x )is a multiple of p since p (x )has degree d −2or less.Therefore,D -nice symmetric polynomials of these types need to be considered separately since formulas (3.2)-(3.4)cannot be used to construct these.But we do not need to derive a formula for constructing D -nice symmetric polynomials of these types,as we will show in Theorem 4.1below.Then we will show formulas are required for the construction of D -nice symmetric polynomials with four roots only when none of the numbers m 1,m 2,and d =2m 1+2m 2are multiples of p .Theorem 4.1.[Results for the cases where D has characteristic p >2and exactly one of m 1,m 2,and d =2m 1+2m 2is a multiple of p .]Let D be an integral domain of characteristic p >2.Suppose p (x )=(x 2−r 21)m 1(x 2−r 22)m 2∈D [x ]has degree d =2m 1+2m 2,and suppose exactly one of the numbers d ,m 1,and m 2is a multiple of p .Then p (x )is D-nice iﬀp (x )splits in D [x ]and p (x )=0.Proof.To prove this result,we consider three cases.In each case,we diﬀerentiate p (x )by the product rule and simplify the derivative.In all cases,it is easy to see that p (x )splits in D [x ]if p (x )=0.Case 1.m 1is a multiple of p ,and m 2and d are not.Then p (x )=2m 2x (x 2−r 21)m 1(x 2−r 22)m 2−1,which always splits in D [x ]since p (x )is never 0.Case 2.m 2is a multiple of p ,and m 1and d are not.Then p (x )=2m 1x (x 2−r 21)m 1−1(x 2−r 22)m 2,which always splits in D [x ]since p (x )is never 0.Case 3.d is a multiple of p ,but m 1and m 2are not.Then p (x )=(−2m 1r 22−2m 2r 21)x (x 2−r 21)m 1−1(x 2−r 22)m 2−1,splits in D [x ]if m 1r 22=m 2r 21.Otherwise,p (x )=0. Remarks.(1).If more than one of the numbers m 1,m 2,and d =2m 1+2m 2is a multiple of p ,then p (x )=0.By our deﬁnition,any polynomial p (x )∈D [x ]where p (x )=0is not D -nice.(2).By Theorem 4.1and its proof,if exactly one of the numbers m 1,m 2,d =2m 1+2m 2is a multiple of p ,then we may construct D -nice symmetric polynomials of these types simply by choosing polynomials that split in D [x ]if d is not a multiple of p .If d is a multiple of p ,we choose polynomials p (x )=(x 2−r 21)m 1(x 2−r 22)m 2that split in D [x ]where m 1r 22=m 2r 21,which is equivalent to the condition that r 2/r 1=± m 2/m 1.It is easy to see that if m 2/m 1is not a square in QF(D ),then p (x )is never 0.Thus,the problem of p (x )=0where exactly one of m 1,m 2,andD-NICE SYMMETRIC POLYNOMIALS WITH FOUR ROOTS219d=2m1+2m2is a multiple of p arises only when d is a multiple of p and m2/m1 is a square in QF(D).A polynomial p(x)=(x2−r21)m1(x2−r22)m2∈D[x]that splits where none of the numbers m1,m2,and d=2m1+2m2are multiples of p is not necessarily D-nice. For example,p(x)=(x2−12)(x2−62)splits in Z/(7)[x],but p(x)is not Z/(7)-nice. So we need formulas(3.2)-(3.4)to construct D-nice symmetric polynomials of these types.Theorem4.2.[Extension of Theorem3.2and Proposition3.5to the case where none of m1,m2,d=2m1+2m2are multiples of p.]Suppose D is an integral domain of characteristic p>2and QF(D)is itsﬁeld of fractions.Suppose p(x)= (x2−r21)m1(x2−r22)m2∈QF(D)[x]has degree d=2m1+2m2,and suppose none of m1,m2,and d are multiples of p.Then Theorem3.2and Proposition3.5apply to p(x).Proof.The proof is the same as the proofs of Theorem3.2and Proposition3.5.We now give three more examples we have found using formulas(3.2)-(3.4).Example4.3.Let D=Z/(11),m1=4,and m2=3.Choose a=2and b=3. Then the formulas give c=5,r1=3,and r2=2.So our D-nice symmetric polynomial isp(x)=(x2−32)4(x2−22)3,(4.1)p (x)=3x(x2−32)3(x2−22)2(x2−52).Example4.4.Let D=Z/(7),m1=2,and m2=1.Choose b=1,and a=2. Then the formulas give c=6,r1=4,and r2=5.So our D-nice symmetric polynomial isp(x)=(x2−42)2(x2−52),(4.2)p (x)=6x(x2−42)(x2−62).Example4.5.Let D=Z/(5)[y],m1=m2=1.Choose a=4y4+3y3+y2+y+2, and b=y3+4y+1.Then the formulas give c=(4y4+3y3+y2+y+2)2+(y3+4y+1)22(4y4+3y3+y2+y+2)+2(y3+4y+1) r1=c−(4y4+3y3+y2+y+2),and r2=c−(y3+4y+1).Stretching our polynomial horizontally by a factor of2(4y4+3y3+y2+y+2)+2(y3+4y+1),。